Function Notation

Many people seem puzzled by function notation well after it has been introduced, and ask “why can’t we just write y = 3x instead of f(x) = 3x ?”. To motivate the use of function notation and improve understanding, I advocate using multi-variable functions instead of single-variable functions in introducing this notation. My introduction usually proceeds something like this:

A Problem

Suppose you are talking to a friend over the telephone, and they have a piece of paper in front of them on which they have drawn a circle using a ruler and a compass. You need the clearest, most concise instructions possible so that you can exactly duplicate the appearance of the paper your friend has… your circle must be exactly the same size as theirs, and in the exact same location on the piece of paper. What information do you need in order to be able to do this?

For the size of the circle, either the radius or the diameter will work nicely (most students come up with this answer quickly). If you are using a compass to draw the circle, the radius will probably be more convenient. However, the radius alone does not tell you where to locate the circle on the paper. What aspect of a circle will do the best job of telling you where it is located (this question is often challenging for students to answer initially)?

The center of a circle is the most useful attribute for identifying its location. The location of a circle’s center on a piece of paper can be described using a coordinate pair (assuming the bottom left corner of the paper is the origin, and the coordinates are given in centimeters).

So, you will only need three numbers to duplicate the circle. Suppose your friend tells you three numbers: “3, 2, and 1”. Can you duplicate their circle with that information? Why, or why not?

There is a small problem… your friend has not told you what each of the numbers represents. This is the first big reason for using “function notation”. If someone asks you to draw

Circle(3, 2, 1)

without giving you any other information, you don’t have enough information to be certain you are drawing the correct circle. Which of the three numbers is the radius? Or is it a diameter? Which is the x-coordinate of the center? Etc.

A Solution

What is missing is the “function definition”. The function definition tells you what the function does, how many parameters (or “arguments”) the function requires as inputs, what those parameters are, and in what order they occur. In this example, the following information is probably what you wanted to know:

Circle(x, y, R) draws a circle with center at (x, y) and radius R

Now you have enough information to interpret “Circle(3, 2, 1)” correctly. The definition tells you the name of the function (for when it is referred to later), how many parameters have to be supplied (three in this case), the order they appear in the parameter list that must be used when the function is invoked (x-coordinate, then y-coordinate, then Radius), and what the function will do with that information.

Function Notation in Math Class

Turning to how function notation will typically be used in math class, you will usually see function definitions that look like this:


Or, in English: there is a function called “f” (instead of “Circle” as used in my example above) which requires only one parameter, which is labelled “x”. You will be asked to evaluate the function “f” for a particular value of “x” using this notation:


This can be read as “what is the value of “f” when its parameter is 4?”. To answer such a question, refer to the right hand side of the function definition above, which tells you to multiply the first (and only) parameter by three, then add two. So, in this example


Furthermore (thanks to Christopher for pointing this out) when you begin to talk about more than one function at a time, as in “compare these two graphs” or “which function reaches 100 first?”, it becomes very useful to have a name for each graph or function. If we had defined them both using “y =”, we would be stuck trying to distinguish between them using potentially ambiguous phrases like “the first equation” or “the curve with the highest y-intercept”.

If equations are defined using function notation, for example


the notation provides names for us to use when referring to them. We can then use “f” and “g” to label each on the graph, refer to them in conversation, and/or invoke them on specific values. This eliminates much confusion when comparing multiple functions, talking about functions of functions such as f(g(x)), or referring to functions defined previously.

Composition of Functions

There is one additional notation you should know about: the symbol used to indicate “composition of functions”, or “taking a function of a function”. The following two expressions are equivalent:

f(g(x))~=~f \circ g(x)

So, when you see the open circle between two function names, evaluate the one to the right of the circle, then plug its result in for the variable in the function on the left of the circle.  Or, if you prefer the notation on the left of the equal sign above, you are welcome to convert the expression to that notation.

For example, using the definitions of “f” and “g” from above:

f \circ g(3)\\*~\\*=~f(g(3))\\*~\\*=~f((3)^2-5)\\*~\\*=~f(4)\\*~\\*=~3(4)+2\\*~\\*=~14

Alternatives to Function Notation

What would we do if function notation did not exist? We would have to invent one or more alternative notations. Yet, we already use such notations all the time.

A single (unary) minus sign tells us to negate a quantity. If we were to define it using function notation, it might look like:

Neg(x) = the value which, when added to x, will equal zero

Instead of using the plus sign to represent addition, we could define the Add function:

Add(x,y) = the arithmetic sum of x and y

Instead of using exponent notation, we could define the Square function as:

Square(x) = the result of multiplying x by itself

However, the notation we use for the most frequently used operations is much more convenient than function notation. Consider for a second how polynomial functions would look in the absence of exponential and arithmetic notation:

Subtract(Add(Multiply(3, Square(x)), Multiply(5,x)),2)\\*~\\*=~3x^2+5x-2

So, function notation is wonderful for complex or infrequently used functions. However, it can get in the way for simple operations or ones that are used frequently.

Function Notation in other subjects

There is one more wrinkle that is useful to know about function notation. You may occasionally see function definitions such as:

Samplefunction(x,y,z)~=~\dfrac{(a)(3x + 2y)}{(z - 4b)}

If you read through the above carefully, you will notice that there are two variables in the definition that are NOT shown in the parameter list. Shock and dismay! What to do? This situation usually occurs when there are constants involved (“a” and “b” in the above example) which people prefer to identify by name instead of by their value. For example, the constant \pi (pi) is almost always referred to by its Greek letter instead of its decimal approximation, because this allows the person using the constant to fill in as many decimal places as they need in their calculation, and also because this gives someone reading the function definition a better idea of what this constant represents (where it came from).

Any variables in a function definition that are NOT included in the parameter list should be constants that you already know from the context in which the function is being used (Physics and Economics are two areas where this happens quite often).

Why is it f(x) so often?

Why don’t math teachers give their functions more interesting names, instead of “f”, “g”, or perhaps even the exotic “h”? Usually because they would like to be able to write the function name quickly and clearly, and because the function is an abstract one… it does not actually correspond to a real-world situation. For example, a function that describes the height of an object above the ground (a projectile motion problem) might be defined as follows by a math teacher who is not discussing any of the physics behind it:


and as follows by a physics teacher

H(t)~=~\frac{\text{-}g }{2}t^2+V_it+ H_i\\*~\\*V_i=10,~~~H_i=4

where “g” is a constant used to represent the acceleration due to gravity, “V_i” is the initial velocity, and “H_i” is the initial height. The initial velocity and height values might be included as variables in the definition in order to remind all students of the role those constants play in the definition, even though their values are given immediately below. A teacher who is going to have to write the name of the function on the board many times over the next few minutes is much more likely to pick a single letter function name (like “H” above, which can be written very quickly and clearly) than a word (like “Height”, as might be used in a computer programming language).

But why is f(x) used so often? I think mostly because “f” is the first letter of the word “function”, and when a teacher is making up a random function to illustrate a point, they do not wish to spend too much time being creative with the name of the function, thus “f” leaps onto the board. So, the next time you see


you may still roll your eyes at the use of “f(x)” instead of “y”, but you will hopefully now understand why this notation exists, what sorts of situations require it to be used, and how to interpret it. From a selfish perspective, I would much rather write the above definition, then ask you to evaluate


than have to write out

…Please calculate the value of “y” when “x” is 5…

Notation is a shorthand, a concise way of representing relationships or operations. Mathematics has many notations, including the arithmetic operators (plus, minus, times, divided by), powers, roots, logarithms, sigma notation, pi notation, and others. Function notation is yet another notation, one which you will see used many times in quantitative subjects like biology, chemistry, physics, economics, probability, statistics, and mathematics.

For additional topics related to Functions, see the following posts:
Function Translations
Function Dilations
Domain and Range

By Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.


  1. Good example with the circle function! I often used a similar example of F(x,y,z) = Temp, which works for high school kids who know where the good/bad seats are in a classroom based on the windows and airvents.

    I struggled with this at the middle school level mainly because *I* like f(x) notation better, but it’s pretty abstract for 7th and 8th. The best results I’ve ever had was when I could introduce it along with some programing – normally Lego, Scratch or Python — that presents systems that follow g(f(x)) structure, even if it’s not explicitly written out that way. I don’t expect it to become their preferred mode of expression, but I want to establish early that input and output are shifting conditional descriptors, and not absolutes tied to the letters x and y.

    1. I agree it is much easier to introduce function notation with programming – that is where my use of the Circle function came from… and it seems to work well for students both because you do NOT have to know any programming whatsoever in order to follow the example, and it is also a function that actually exists on TI graphing calculators – so they can try it out (I just checked out the the TI-83 manual, and the parameters are Circle(x,y,Radius) – so I guess I’ll modify my posting to conform to that – oops).

      I also agree that x and y are way overused (we all fall into habits), and it is a great idea to mix it up whenever possible. Once again, if students are programming in a language that supports functions that return a value (not in one of the parameters), and variables become words instead of letters, it can provide first hand experience that helps it all make a great deal more sense.

      1. A coordinate point is usually thought of as having two elements: (x,y)

        The first element of a coordinate point is often referred to as the “input”, “independent variable”, or “domain”. When using function notation, the item(s) in parentheses after the function name are the inputs to the function… so it becomes the leftmost element of the coordinate point.

        The second element of a coordinate point is often referred to as the “output”, “dependent variable”, or “range”. The result of evaluating a function of a some value is the “output” of the function, so that becomes the rightmost element of the coordinate point.

        So, if:
        f(x) = 2x+3
        f(5) = 13
        which means, for an input value of 5, the function produces a result of 13. Therefore, as a coordinate point, this would be
        (5, 13)
        which can be interpreted as: an input value of 5 is associated with an output value of 13.

  2. I had this very conversation today with an adult student in the Math Center at my college. Her question, “Why are they using f(x)? and where does g(x) come from?”
    I’ll have to think about your CIRCLE example and how to add it to the mix. If I’m a student with this question, I think I’ll be convinced that we need three variables to define the circle function, but I might still wonder why I can’t just write y=5x+12.
    My own approach has been to give permission at the early stages to mentally substitute “y=” whenever we see “f(x)=”, but that there are reasons we need the new notation. The first of these is to be able to work with two functions at the same time; if we call them both “y” then we can’t sort out which is which. Function notation allows us to name each separately so we can refer to “f(x)” and “g(x)” unambiguously. The other reason we need function notation is that it allows us to do things that y= notation does not-especially composition. There is no equivalent “y=” way of expressing f(g(x)).
    I’ll be thinking about what the circle example could add to the mix. Thanks for the post.

    1. Christopher,

      Good point about needing to work with two functions at the same time without causing confusion – I will add it into my post!

      I have no qualms about students using “y=” when it is unambiguous. I see Function Notation as something that becomes useful only when we need it (how is that for a department of redundancy department statement?).


  3. Great article.

    I was working with an adult learner today, on functions. They were fine until something like f(x) = 2x + 5, and then they had to evaluate something like f(2x+5)

    Any ideas how to provide motivation for the usefulness of that?

    They understood the principle fine, but really couldn’t see any point in doing those kinds of replacements.

    Once again, good article!


    1. Tom,

      My favorite approach to substitution is to recommend:

      1) Rewrite the function definition, but use a gaping set of parentheses for each place that a variable appears. So, for your example it would become:

      2) Now place the function argument, exactly as it was given, no matter how ugly, into the appropriate set(s) of parentheses in your function definition. So, for your example it would now become:

      3) And finally, simplify:

      As to why this would be useful… two uses come to mind:
      1) When creating horizontal translations or dilations we are asking students to understand how the graph of f(2x) or f(x+3) relate to the graph of f(x).
      2) In an abstract sense, we are preparing students to “nest” or “compose” functions: f(g(x)). This has direct parallels to computer programming, where one subroutine can call another, which calls another, etc. That is the most relevant real-world example from my experience, and it corresponds to the idea that many problems can be more easily solved by breaking them down into simpler components.

      So, for example, a complicated solution to a particular mathematical challenge could be broken down into a series of multi-step phases, each of which makes decent sense on its own. The first solution phase could be described by some function g(x). The next phase would involve taking the answer from the first phase, and manipulating it in a series of ways as described by h(x). So, to go straight from problem to solution, the complete solution now becomes described by h(g(x)).

      Taken as a whole, the entire solution might be difficult to understand. However, individually, each of the steps h(x) and g(x) are comprehensible.

      Does that help?

      1. I’d like to weigh in on this, too. Whit’s ideas about decomposing complicated functions into manageable pieces (and then recomposing them) is a nice way of thinking about composition outside of mathematics.
        But I want to put in a plug for the pure mathematical explanation, too. Mathematics is a beautiful subject even (especially?) when not applied to other things. There are a number of basic questions at the heart of what it means to do mathematics. One of these is “What if we treat this new mathematical object like we do the older, more familiar ones?” That’s what is happening when we compose functions.
        Functions are made up of numbers and variables (f(x)=3x+2). We are used to treating those like each other by adding, multiplying, etc. And with function notation, we are used to writing f(4)=3*4+2-substituting a number for a variable. But what if we treated a function like a variable? Does that make sense? Can we substitute a function for x? What new things can we do mathematically if allow this possibility? When are you ever going to USE this? Maybe if you go into computer science, as Whit suggests. But we don’t use poetry very often in our jobs, so why must it be different with beautiful mathematical ideas?
        So I say, give both motivations-the practical programming one and give a sales pitch on the beauty of the idea of “extension” in mathematics-treating new objects (here functions) like old ones (here variables).

      2. Christopher asks “But what if we treated a function like a variable?”. Another way of phrasing it could be: “But what if a variable was itself a function of some other variable(s)?” What if there is another layer of understanding the lies beneath the one we have just explored?

        This idea can be used to explain things in the context of a “parent function”, like vertical translations:
        f(x) = x^2\\*UpBy5(x) = x + 5\\*UpBy5(f(x)) = x^2 + 5

        and horizontal translations:
        f(x) = x^2\\*RightBy5(x) = x - 5\\*f(RightBy5(x)) = (x-5)^2

        and both put together:
        UpBy5(f(RightBy5(x))) = (x-5)^2+5

  4. mmm its good and i understand a lot of this … function .. for the medical students this is must be so easy 🙂 thnx

  5. Thanks. Really helpful. The question i want to know is why would we not use Function notation in certain situations?

    1. Nice question, Roman. I find myself NOT wanting to use function notation when I want to solve for the dependent variable. If I write f(x)=x^2, it is somehow less satisfying to solve as x=sqrt(f(x)) than it would be if I had written y=x^2 and solved as x=sqrt(y).

    2. Roman,

      That’s a good question!

      1) I think function notation is NOT used in early Algebra (or earlier) because of concern that it could confuse students, and the notation is not really needed yet. The mathematical expression we are working with are fairly simple, and do not benefit from being hidden behind a name.

      2) Once the notation becomes useful and needed (when we need to be talking about complex or messy expressions, multiple functions at once, multivariate functions, or we need to be clear about what each function/equation uses as an independent variable), it is worth taking the time to introduce the notation and clear up any confusions with the notation because it becomes very useful.

      Using a function name to refer to a complex relationship introduces a level of abstraction to the problem (which students are now hopefully cognitively ready to deal with) and can make it easier to describe and understand complex situations.

      3) However… to finally answer your question, there are times when function notation is not useful. For example, when describing a quantitative relationship, sometimes you end up writing two expressions that are related to one another in some way (=, >, <). Multiple variables may be involved, nothing has been "solved for". At this stage of thinking, function notation is not necessarily useful.

      You could be setting the results of two different functions equal to one another, at which point you might need/wish to substitute the function definitions in for the function notation in order to work the problem further. So function notation might disappear from the problem at this point, and the solution to the problem might not be one that needs function notation to help explain.

      4) And then there are the in-between situations. For example, linear equations are not usually taught using function notation, although they certainly could be. Why? Because writing "f(x)=" takes more ink/marker/chalk than writing "y=", and does not necessarily add any clarity when students are grappling with the basics of plotting points, understanding the interpretation of intercepts and slope, etc.

      Notation is a convenience. It is usually used when it allows us to express things more clearly and/or concisely. If it does not achieve that objective, then it is not used. For example, using an "x" to indicate multiplication is very useful for younger students who think more concretely, and might not perceive "xy" as a product, since there is no arithmetic operator written between the two variables. So using "x" to indicate multiplication is a very useful notation with younger students. However, as their abstract thinking skills grow, we abandon that notation because it begins to get in the way (does "x" represent multiplication or the variable? Writing the "x" takes more time/ink than just writing the two variable symbols next to each other, etc.).

      We abandon the "x" pretty close to the time that we also introduce the use of function notation, because one has outlived its usefulness, while the other is just beginning to be useful. As the quantitative relationships we are describing grow in complexity, it become more and more useful to be able to "hide" portions of the expression by using function notation instead of the full function definition. This can make it easier to "take in" the entire description, and function notation is therefore serving a purpose that would have been more difficult to achieve without it.

      I have probably missed some other good examples or analogies – but hopefully the above gives you the beginnings of an answer to your question.

    1. While Math (and many other subject) topics can feel difficult when you first learn them, you will understand them better the more you work with them. Be patient with yourself, keep trying, and you will find that it all makes sense in time. All of your courses will get more challenging every year, and you will discover that you rise to the challenge. Many skills you learn in Math can be useful in many areas of your life… see this post.

  6. Hello. I am an adult who is interested in learning more about math. I did not take math courses in college and was a foreign language major. For a while I’ve been having trouble distinguishing between functions and equations, and why we need different notation for both. Recently I came across the discussion on this page in which this explanation is given:

    “Furthermore (thanks to Christopher for pointing this out) when you begin to talk about more than one function at a time, as in “compare these two graphs” or “which function reaches 100 first?”, it becomes very useful to have a name for each graph or function. If we had defined them both using “y =”, we would be stuck trying to distinguish between them using potentially ambiguous phrases like “the first equation” or “the curve with the highest y-intercept”.

    My first question is why can’t we just say “y=3x+2” for the first equation and “z=x2 – 5” for the second one instead of f(x)= and g(x)= ?

    Another question I have is about why we need function notation to do composition. For example, if a store has a discount sale on shirts (I got this example from the web) where all shirts are 20% off, and preferred customers get an additional 15% off of that price, then we could write two functions:

    1) S(C) = .8C
    where S(c) is the function for the Sale price, and “C” stands for regular price before applying the 20% sale discount

    2) P(C)= .85C
    where P(C) is the function for the Preferred Customer price, and “C” stands for the price before applying the preferred customer discount of 15%

    To find out what preferred customers pay we could compose the two functions and say:
    P(S(C)) = .85(S(C)) = .85(.8C) = .68C

    But why couldn’t we just say for the first function that “S=.8C” (where S stands for 20% off Sale price), and for the second one “P=.85S (where P = Preferred Customer price)? This took two steps instead of three.

    Thank you!

    1. Joseph,

      1) You are right, using different dependent variables would indeed differentiate the two equations successfully. You would then either graph the two equations on separate sets of axes (one x-y and the other x-z), or you would leave the vertical axis unlabeled and label each line with its dependent variable (just as you would with function names).

      Sometimes, using different variables can imply different quantities being calculated. For example, H is height, D is distance, W is weight, etc. so using two different dependent variables could imply that different types of quantities are being calculated, which might call into question whether it is appropriate to graph them on the same vertical axis. At other times, this would not be an issue, for example: J is John’s height, B is Bill’s height.

      Why use function notation? Because it summarizes critical information nicely, particularly in multivariate situations such as the Circle function I describe. It allows the reader to quickly determine how many parameters are needed, and in what order they will be supplied, so that the function can be evaluated correctly. This is very useful when working in multivariate situations or programming computers, where many functions require more than one argument. For example, combinations can be described more clearly and succinctly for many using function notation C(5,2) instead of writing out the formula explicitly _nC_r=\frac{n!}{(n-r)!r!}=\dfrac{5!}{3!2!}.

      However, this is not as useful when working with linear equations that have only one independent variable. So, function notation becomes increasingly useful when working with complex situations, but does not seem to have as much obvious value when working with simpler situations.

      2) You are right again. Function notation is not necessary to compose two functions. You are welcome to substitute one into the other as you did in your example.

      However, function notation does becomes useful when seeking to talk and think about the idea of composing generic functions… f composed on g, versus g composed on f. Under what circumstances would the two orders produce the same result, and when would they produce different results? Are there general principles we can arrive at here?

      Mathematics and algebra are abstract topics. They seek to find “truths” about specific types of situations that can be used to improve our understanding and/or ability to answer questions about them. So, while math often starts with concrete or realistic situations from the world around us, it usually gravitates quickly to abstractions about those situations that are intended to help us figure out how to represent them algebraically. Once represented using the notations of algebra, we can (hopefully) use algebraic procedures to answer questions about the situation efficiently and confidently.

      By using two specific functions, and substituting one into the other as you did in your example, the reader is more likely (in my view) to be thinking about the specifics of the situation you are modeling and less likely to be thinking abstractly about the patterns involved. By using function notation to represent abstract or unspecified functions, it is easier to get the reader to begin thinking abstractly, to begin looking for valid generalizations that might apply to all situations of a certain type.

      Many examples given in textbooks, particularly examples cited just as a new notation is being introduced, are simple ones. The simplicity is intended to make the situation easier to understand, but with function notation I think it has the reverse effect… it makes the reader wonder why the notation is being introduced at all. If more complex examples were used (such as the Circle function I cite), I think the relevance of the notation would become a bit more clear. Or, the notation could be introduced later… when it is desperately needed to remove some of the visual clutter from a situation. I think texts introduce it when they do for several reasons: because colleges assume students will be familiar with it, and because it is useful to know for the SAT.

      I suspect that you will become comfortable with using function notation fairly quickly, and its value will become more apparent as you work with increasingly complex situations over time. For now, what is most important is that you know how to interpret it when you see it, or use it if you are called upon to do so. Yes, there are other ways of accomplishing the same tasks that function notation accomplishes – and you need to be familiar with and confident when using all of them… just as you do with the various notations used to represent multiplication:
      X\times Y\\*X\cdot Y\\*XY\\*(X)(Y)

      1. Also, sorry I didn’t respond sooner. I did not get email notification about your reply, but I just tried to fix my settings so that in the future I can get email notifications.

        Regarding functions getting easier as I get into more complex math, I look forward to learning more in the near future. Right now I’m working on trigonometry and have been reading some of the basic ideas behind calculus which I found very interesting.

        Thanks again.


      2. Trig and calculus are both very interesting and useful topics. Trig in particular is very useful in modeling all sorts of things (tides, rise and fall of the sun, annual temperatures, body temperatures over a 24 hour period, etc.) Trig functions also introduce you to all possible translations and dilations – so I hope you find my posts on those topics of interest when the time comes.

        Best of luck, and I hope you enjoy your mathematical excursions! While mastering some topics may take considerable time, thought, and practice, I find the insights achieved usually feel “worth the time” needed when all is said and done.

    1. Augustus,

      I do not recall seeing that notation before.

      It could be a typo, where a semi-colon was typed instead of a comma. If that is the case, then f is a function of four variables: t, x, y, and z. In other words, the value of f depends on both 3-dimensional position and time. This would be my best guess scenario.

      It could be a different typo, where a semi-colon was typed instead of a colon. If that is the case, I suppose it could be interpreted as something like f is a function of t, which is a 3-dimensional vector with endpoint coordinates of x, y, and z. However, if t represents time, time is not a vector quantity in the physics I am familiar with (maybe this changes in more advanced physics).

      If you can provide a bit more context for the notation, that might generate some better ideas…

  7. The idea behind the notation, after having discussed it with the author, is that over small intervals of time (small is not defined), changes in x, y, and z are considered negligible with respect to time but the context in which the function is evaluated with respect to changes in x,y, and z ignoring how they change with respect to t but not ignoring the fact that changes in the function are not negligible with respect to small changes in time (small interval). Hope this is understandable.

    1. Augustus,

      Thank you for closing the loop! I hate leaving uncertainty unresolved…

      Is it fair to paraphrase your description by saying: the notation f(t; x, y, z) means that f is “primarily” a function of t, but is also depends on x, y, and z. This emphasis on t is used because the output of the function is quite sensitive to changes in t over small intervals of time, but much less sensitive to changes in x, y, or z over small intervals of time.

      So – it sounds like you are probably in a calculus-based course, analyzing how a function behaves in response to changes in each of its various inputs. In this particular example, the emphasis on t makes the analysis simpler, since you can focus on df/dt without considering the other partial derivatives (as long as you are considering a narrow range of values for t, i.e. a sufficiently small time interval).

      Is that consistent with what you were told?

      And thanks again for pursuing an answer and sharing it!

      1. Whit,

        Actually, I have been in radar technology (undergraduate in electrical
        engineering) for many years. I did my graduate work in Applied
        Mathematics and PhD in electrical engineering. The notation was new to me
        so I wanted to be sure nothing had come along to alter my knowledge.
        However, I am working on extending the Taylor Series Expansion to an
        engineering application with the person (author) who was responsible for
        the notation. (I wanted an independent opinion from someone other than
        the author.) Hence, partial derivatives of f with respect respect to all
        four variables since changes are extremely important in the application
        because an assessment must be made with respect to incremental changes in
        x,y, and z but not so much with t. But there is one other aspect to
        consider. You stated, “This emphasis on t is used because the output of
        the function is quite sensitive to changes in t over small intervals of
        time.” Just change small to not small or whatever word, eg large, is more

        Your comments are sincerely appreciated and I hope this clarifies the



    1. Mark,

      In general f(x+y) will not usually equal f(x) + f(y). However, this could be true for some functions, or for some particular values of x and y for some functions (and not others)… so you need to test whether this is true or not for this particular function using the values of 3 and 1:

      Evaluate f(3+1)… what does it equal?

      Evaluate f(3), then evaluate f(1), and add the two together.

      You now have enough information to answer the question: do the two equal each other, or not?

      An interesting extension of this problem is to figure out whether this will always be true for f(x), or is only true for 3 and 1:

      Evaluate f(x+y)… you will end up with an expression involving both x and y.

      Evaluate f(x), evaluate f(y), and add the two together.

      Are your two expressions, both involving x and y, equal to each other when simplified? If so, then this would always be true for this function. If not, then this may only true for certain values of x and y (if any).

  8. Consider revising your input values for the circle please. If you actually try and draw this circle using your parameters, the left side of the circle hangs off the page. Given an x value of only 3 with a radius of 7 causes this problem. Consider reversing these values to make students more clearly see the results when attempting. I understand the point is function notation, however, students may get hung up on their picture of the circle not fitting onto the page and think they misunderstood your method. Thanks.

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