0. Functions

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

ƒ(ƒ(h(3)))

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>

c. ƒ(g(-3))

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

g(ƒ(4))

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

(g o ƒ ) (x) = x⁴ + 3

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

e. g(f(0))

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

ƒ(x) = (x)/(x+1)

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.

f. y = √(x³ − 3)

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).

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

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)

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) = 1/x², x > 0

Suppose ƒ is an even function with ƒ(2) = 2 and g is an odd function with g(2) = -2. Evaluate ƒ(-2) , ƒ(g(2)), and g(ƒ(-2))

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

ƒ(x) = 2x² -3x +1

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

f. f(g(1/2))

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?

g. g ∘ f