Composition of Functions:
Composing Functions with Functions
(page 3 of 6)

Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition

You can also evaluate compositions symbolically. It is simpler to evaluate a composition at a point because you can simplify as you go, since you'll always just be plugging in numbers and simplifying. Evaluating a symbolic compositon, where you're first plugging x into some function and then plugging that function into some other function, can be much messier. But the process works just as the at-a-number composition does, and using parentheses to be carefully explicit at each step will be even more helpful.

• Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( f o g)(x).
• In this case, I am not trying to find a certain numerical value. Instead, I am trying to find the formula that results from plugging the formula for g(x) into the formula for f(x). I will write the formulas at each step, using parentheses to indicate where the inputs should go:

( f o g)(x) = f (g(x))
= f (–x2 + 5)
= 2(             ) + 3     ... setting up to insert the input formula
= 2(–x2 + 5) + 3
= –2x2 + 10 + 3
= –2x2 + 13

If you plug in "1" for the x in the above, you will get ( f o g)(1) = –2(1)2 + 13 = –2 + 13 = 11, which is the same answer we got before. Previously, we'd plugged a number into g(x), found a new value, plugged that value into f(x), and simplified the result. This time, we plugged a formula into f(x), simplified the formula, plugged the same number in as before, and simplified the result. The final numerical answers were the same. If you've done the symbolic composition (the composition with the formulas) correctly, you'll get the same values either way, regardless of the value you pick for x. This can be a handy way of checking your work.

• Given f(x) = 2x + 3 and g(x) = –x2 + 5, find (g o f )(x).
• (g o f )(x) = g( f(x))
= g(2x + 3)
= –(           )2 + 5    ... setting up to insert the input
= –(2x + 3)2 + 5
= –(4x2 + 12x + 9) + 5
= –4x2 – 12x – 9 + 5
= –4x2 – 12x – 4

There is something you should note from these two symbolic examples. Look at the results I got:

( f o g)(x) = –2x2 + 13
(g o f )(x) = –4x2 – 12x – 4

That is, ( f o g)(x) is not the same as (g o f )(x). This is true in general; you should assume that the compositions ( f o g)(x) and (g o f )(x) are going to be different. In particular, composition is not the same thing as multiplication. The open dot "o" is not the same as a multiplication dot "•", nor does it mean the same thing. While the following is true:

f(x) g(x) = g(x) f(x)            [always true for multiplication]

...you cannot say that:

( f o g)(x) = (g o f )(x)           [generally false for composition]

That is, you cannot reverse the order in composition and expect to end up with the correct result. Composition is not flexible like multiplication, and is an entirely different process. Do not try to multiply functions when you are supposed to be plugging them into each other.

You can use the Mathway widget below to practice function composition. Try the entered exercise, or type in your own exercise. Then click the "paper-airplane" button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)

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 Cite this article as: Stapel, Elizabeth. "Composing Functions with Functions." Purplemath. Available from     http://www.purplemath.com/modules/fcncomp3.htm. Accessed [Date] [Month] 2016

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