0. Functions

In Exercises 41 and 42, (a) write formulas for ƒ ○ g and g ○ ƒ and find the (b) domain and (c) range of each.

ƒ(x) = 2 - x², g(x) = √ x + 2

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.

f(x) = (x + b) / (x − 2), b > −2 and constant

Combining Functions

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?

b. f/g

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

c. y = x¹/⁴

Algebraic Combinations

In Exercises 3 and 4, find the domains of f, g, f/g and g/f.

f(x) = 1, g(x) = 1 + √x

Composition of Functions

In Exercises 39 and 40, find

d. (g ○ g) (x).

ƒ(x) = 1/x , g(x) = 1/√ x + 2

In Exercises 41–44:

a. Find f⁻¹(x).

43. f(x) = 5 − 4x, a = 1/2

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.

Find the domain of ƒ o g.

If ƒ(x) = √x and g(x) = x³-2 and , simplify the expressions (ƒ o g) (3), (ƒ o ƒ) (64), (g o ƒ) (x) and (ƒ o g) (x)

Working with composite functions Find possible choices for outer and inner functions ƒ and g such that the given function  h equals ƒ o g .

h(x) = √ (x⁴ + 2 )

Working with composite functions

Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.

h(x) = (2) / ( x⁶ + x² + 1)²

Use the graphs of ƒ and g in the figure to determine the following function values. y = f(x) ; y=g(x) <IMAGE>

c. ƒ(g (4))

Use the table to evaluate the given compositions. <IMAGE>

g(h(ƒ(4)))

Combining Functions

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?

d. f² = ff