0. Functions

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

ƒ(x) = 10

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

ƒ(x) = 4x-3

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

ƒ(x) = 2x² -3x +1

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

ƒ(x) = (x)/(x+1)

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

ƒ(x) = 4 - 4x + x²

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.

ƒ(x) = - (3/√x)

Another way to approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.

c. Explain why it is not necessary to use negative values of h in the table of part (b).

Inverse of composite functions

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

{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 Functions

In Exercises 39 and 40, find

a. (ƒ ○ g) (-1).

ƒ(x) = 1/x , g(x) = 1/√ x + 2

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 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

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

ƒ₁(x)        ƒ₂(x)

 x²          |x|²

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

ƒ₁(x)        ƒ₂(x)

 x³          |x³|

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

ƒ₁(x)        ƒ₂(x)

4 - x²       |4 - x²|