Intermediate Algebra (11th edition)

Let and

Let and.

= 12 x+ 122 = 2 x^2 + 1

=ƒ 12 x+ 12 =g 1 x^22

1 ƒg 21 x 2 =ƒ 1 g 1 x 22 1 gƒ 21 x 2 =g 1 ƒ 1 x 22

ƒ 1 x 2 =x^2 g 1 x 2 = 2 x+ 1

=x^2 - 2 x- 1

=x^2 + 2 x+ 1 =x^2 - 12 x+ 12

=ƒ 1 x 2 +g 1 x 2 =ƒ 1 x 2 - g 1 x 2

1 ƒ+g 21 x 2 1 ƒ-g 21 x 2

ƒ 1 x 2 =x^2 g 1 x 2 = 2 x+ 1.

5.3 Polynomial Functions, Graphs,

and Composition

Adding and Subtracting Functions

If and define functions, then

and

Composition of fand g

Graphs of Basic Polynomial Functions

1 ƒg 21 x 2 ƒ 1 g 1 x 22

1 ƒg 21 x 2 ƒ 1 x 2 g 1 x 2.

1 ƒg 21 x 2 ƒ 1 x 2 g 1 x 2

ƒ 1 x 2 g 1 x 2

x

y

(–1, –1) (0, 0)

(–2, –2)

(1, 1)

(2, 2)

f(x) = x

Identity function

f(x) = x

Domain: (–∞, ∞)

Range: (–∞, ∞)

x

y

(0, 0)

(1, 1)

(2, 4)

f(x) = x^2

(–1, 1)

(–2, 4) Squaring function

f(x) = x^2

Domain: (–∞, ∞)

Range: [0, ∞)

x

y

(0, 0)

(1, 1)

f(x) = x^3 (2, 8)

(–1, –1)

(–2, –8)

Cubing function

f(x) = x^3

Domain: (–∞, ∞)

Range: (–∞, ∞)

5.4 Multiplying Polynomials

To multiply two polynomials, multiply each term of one
by each term of the other.

To multiply two binomials, use the FOIL method.
Multiply the Firstterms, the Outerterms, the Innerterms,
and the Lastterms. Then add these products.

Special Products

1 xy 22 x^2  2 xyy^2

1 xy 22 x^2  2 xyy^2

1 xy 21 xy 2 x^2 y^2

FOIL

= 25 a^2 + 30 ab+ 9 b^2 = 4 k^2 - 4 k+ 1

15 a+ 3 b 22 12 k- 122

= 9 m^2 - 64

13 m+ 8213 m- 82

= 2 x^2 - 11 x- 21

= 2 x^2 - 14 x+ 3 x- 21

= 2 x 1 x 2 + 2 x 1 - 72 + 3 x+ 31 - 72

12 x+ 321 x- 72

= 4 x^5 - 5 x^4 + 14 x^3 - 15 x^2 + 6 x

= 4 x^5 + 12 x^3 - 5 x^4 - 15 x^2 + 2 x^3 + 6 x

1 x^3 + 3 x 214 x^2 - 5 x+ 22

Multiplying Functions
If and define functions, then

1 ƒg 21 x 2 ƒ 1 x 2 #g 1 x 2.

ƒ 1 x 2 g 1 x 2 Let and

= 2 x^3 +x^2

=x^2 12 x+ 12

=ƒ 1 x 2 #g 1 x 2

1 ƒg 21 x 2

ƒ 1 x 2 =x^2 g 1 x 2 = 2 x+ 1.

CONCEPTS EXAMPLES

310 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions

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