0. Functions

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>

g. ƒ (g(g(-2)))

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3). Simplify or evaluate the following expressions.

g(1/z)

Show that the graph of the inverse of f(x)=mx+b, where m and b are constants and m≠0, is a line with slope 1/m and y-intercept -b/m.

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>

a. ƒ(g(-2))

Inverse of composite functions

c. Explain why if g and h are one-to-one, the inverse of ƒ(x) = g(h(x)) exists.

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

ƒ o g

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x)        ƒ₂(x)

 __         ___

√ x         √ |x|

Missing piece Let g(x) = x² + 3 Find a function ƒ  that produces the given composition.

(ƒ o g) (x) = x⁴ + 6x² + 9

Composition of Functions

Copy and complete the following table.

d. <IMAGE>

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>

i. g(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.

F(y⁴)

Composition of Functions

Copy and complete the following table.

a. <IMAGE>

What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?

Find the inverse of f(x)=-x+1. Graph the line y=-x+1 together with the line y=x. At what angle do the lines intersect?

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) = x² − 2x, x ≤ 1