0. Functions

Given the functions $f(x)=x+3$ and $g(x)= x^2$ find $(f∘g)(2)$ and $(g∘f)(2)$.

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.

f(x) = x² - 2x

Composite functions

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

Find ƒ(g(h( x))).

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 4x-3

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = 9x - 8

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 10

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(g(y))

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

(g o ƒ ) (x) = x²⸍³ + 3

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

g(ƒ(h(4)))

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)=x³-1

Composition of polynomials

Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.

What is the degree of the following polynomials?

ƒ o g

In Exercises 41–44:

a. Find f⁻¹(x).

44. f(x) = 2x², x ≥ 0, a = 5