CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 1. Functions and Graphs

What isf

(

j(h(g(x)))

)

?

Solution:These functions arenestedwithin the arguments of the other functions. Sometimes functions simplify
significantly when composed together, asfandjdo in this case. It makes sense to evaluate those two functions first
together and keep them on the outside of the argument.
f(x) =x^2 −1;h(x) =xx+−^15 ;g(x) = 3 ex−x;j(x) =

√

x+ 1

f(j(y)) =f

(√

y+ 1

)

=

(√

y+ 1

) 2

− 1 =y+ 1 − 1 =y

Notice how the composition offandjproduced just the argument itself?
Thus,

f

(

j(h(g(x)))

)

=h(g(x)) =h( 3 ex−x)

=(^3 e

x−x)− 1

( 3 ex−x)+ 5

=^3 e

x−x− 1

3 ex−x+ 5

Concept Problem Revisited
Function composition is not the same as multiplying two functions together. With function composition there is an
outside function and an inside function. Suppose the two functions were doubling and squaring. It is clear just
by looking at the example input of the number 5 that 50 (squaring then doubling) is different from 100 (doubling
then squaring). Both 50 and 100 are examples of function composition, while 250 (five doubled multiplied by five
squared) is an example of the product of two separate functions happening simultaneously.

Vocabulary

Function compositionis when there are two or more functions and the range of the first function becomes the
domain of the second function.
Nestingrefers to a function being operated on or in the argument of another function.
Acounterexampleis a specific instance that contradicts a statement. When you are asked to show a statement is not
true, it is best to find a counterexample to the statement.

Guided Practice

For the three guided practice problems use the following functions.
f(x) =|x|