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

Express the radius of a sphere as a function of the sphere’s surface area. Then express the surface area as a function of the volume.

Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables.

Graph the functions in Exercises 13–22. What is the period of each function?

sin (x/2)

Graph the functions in Exercises 13–22. What is the period of each function?

sin (x − π/4) + 1

Graph the functions in Exercises 23–26 in the ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have?

s = −tan πt

[Technology Exercise]

a. Graph y = cos x and y = sec x together for −3π/2 ≤ x ≤ 3π/2. Comment on the behavior of sec x in relation to the signs and values of cos x.

Graph y = sin x and y = ⌊sin x⌋ together. What are the domain and range of ⌊sin x⌋?

State whether each function is increasing, decreasing, or neither.

c. Height above Earth’s sea level as a function of atmospheric pressure (assumed nonzero)

State whether each function is increasing, decreasing, or neither.

d. Kinetic energy as a function of a particle’s velocity

Find the largest interval on which the given function is increasing.

a. ƒ(x) = |x - 2| + 1

Find the largest interval on which the given function is increasing.

d. R(x) = √ 2x - 1

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|

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|

Shifting and Scaling Graphs

Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.

e. Stretch vertically by a factor of 5

Shifting and Scaling Graphs

Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.

f. Compress horizontally by a factor of 5

For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.

cos 2θ + cos θ = 0

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

a. 𝔂 = ƒ(x - 5)

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

d. 𝔂 = ƒ(2x + 1)

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

e. 𝔂 = ƒ( x ) - 4

3

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.

y = - √(1 + x/2)

Derive a formula for tan (A − B).

Apply the formula for cos (A − B) to the identity sin θ = cos (π/2 − θ) to obtain the addition formula for sin (A + B).

The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then

(sin A) / a = (sin B) / b = (sin C) / c

Use the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.

In Exercises 59–62, sketch the graph of the given function. What is the period of the function?

𝔂 = cos πx/2