0. Functions

Composite functions

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

Find h (ƒ (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) = (x³ - 5)¹⁰

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51). 

b. Consider the polynomial g(x) = f(f(x)). Write g in terms of a and powers of x. What is its degree?

Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

a. ƒ(g(-1))

Composition of Functions

In Exercises 39 and 40, find

a. (ƒ ○ g) (-1).

ƒ(x) = 1/x , g(x) = 1/√ 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) = x⁵

Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

e. g(g(-1))

Composition of even and odd functions from graphs Assume ƒ is an even function and g is an odd function. Use the (incomplete) graphs of ƒ and g  in the figure to determine the following function values. <IMAGE>

e. g(g(-7))

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

a. (ƒ o g ) (2)

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

c. g(g(−1))

Composite functions and notation

Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

ƒ (√(x+4))

Suppose that the range of g lies in the domain of f so that the composition fog is defined. If f and g are one-to-one, can anything be said about fog? Give reasons for your answer.

Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.

f(x)=(x+1)², x≥-1

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

d. g(ƒ(5))

Given the functions $L\(\left\)(x\(\right\))=x-2$ and $M\(\left\)(x\(\right\))=x^2$, calculate $\(\frac{L}{M}\]\left\)(5\(\right\))$