(b)
Use the definition.
Substitute.
Change subtraction to addition.= 4 x^2 + 4 x Add. NOW TRY
= 1 x^2 - 3 x+ 72 + 13 x^2 + 7 x- 72
= 1 x^2 - 3 x+ 72 - 1 - 3 x^2 - 7 x+ 72
= ƒ 1 x 2 - g 1 x 2
1 ƒ- g 21 x 2
286 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions
NOW TRY
EXERCISE 4
For
and ,
find each of the following.
(a)
(b) 1 ƒ-g 21 - 42
1 ƒ+g 21 x 2g 1 x 2 =- 6 x^2ƒ 1 x 2 =x^2 - 4Adding and Subtracting FunctionsFind each of the following for the functions defined by
and
(a) 1 ƒ+ g 2122
ƒ 1 x 2 = 10 x^2 - 2 x g 1 x 2 = 2 x.
EXAMPLE 4
Use the definition.Substitute.
Order of operations= 40 Subtract, and then add.
= 340 - 44 + 4
= 3101222 - 21224 + 2122
f (x)= 10 x^2 - 2 x g(x)= 2 x= ƒ 122 + g 122
This is a
key step.Alternatively, we could first find
Use the definition.
Substitute.
Combine like terms.Then,
Substitute.
The result is the same.(b) and
Use the definition.
Substitute.
Combine like terms.Then,
Substitute.
Simplify.Confirm that ƒ 112 - g 112 gives the same result. NOW TRY
= 6.
= 101122 - 4112
1 ƒ- g 2112
= 10 x^2 - 4 x
= 110 x^2 - 2 x 2 - 2 x
= ƒ 1 x 2 - g 1 x 2
1 ƒ- g 21 x 2
1 ƒ- g 21 x 2 1 ƒ-g 2112
= 40.
= 101222
1 ƒ+g 2122
= 10 x^2
= 110 x^2 - 2 x 2 + 2 x
= ƒ 1 x 2 +g 1 x 2
1 ƒ+ g 21 x 2
1 ƒ+g 21 x 2.
OBJECTIVE 4 Find the composition of functions.The diagram in FIGURE 1on
the next page shows a function ƒ that assigns, to each element xof set X, some ele-
ment yof set Y. Suppose that a function gtakes each element of set Yand assigns a
value zof set Z. Then ƒ and gtogether assign an element xin Xto an element zin Z.
The result of this process is a new function hthat takes an element xin Xand assigns
it an element zin Z.
NOW TRY
EXERCISE 3
For
and ,
find each of the following.
(a)
(b) 1 ƒ-g 21 x 2
1 ƒ+g 21 x 2g 1 x 2 =- 2 x^3 +x^2 - 12ƒ 1 x 2 =x^3 - 3 x^2 + 4NOW TRY ANSWERS
- (a)
(b) - (a)
(b) 108- 5 x^2 - 4
3 x^3 - 4 x^2 + 16- x^3 - 2 x^2 - 8