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문제 1.2.79g
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
문제 2
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.
문제 5
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.
문제 14
Graph the functions in Exercises 13–22. What is the period of each function?
sin (x/2)
문제 21
Graph the functions in Exercises 13–22. What is the period of each function?
sin (x − π/4) + 1
문제 24
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
문제 27a
[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.
문제 29
Graph y = sin x and y = ⌊sin x⌋ together. What are the domain and range of ⌊sin x⌋?
문제 33c
State whether each function is increasing, decreasing, or neither.
c. Height above Earth’s sea level as a function of atmospheric pressure (assumed nonzero)
문제 33d
State whether each function is increasing, decreasing, or neither.
d. Kinetic energy as a function of a particle’s velocity
문제 34a
Find the largest interval on which the given function is increasing.
a. ƒ(x) = |x - 2| + 1
문제 34d
Find the largest interval on which the given function is increasing.
d. R(x) = √ 2x - 1
문제 39a
Composition of Functions
In Exercises 39 and 40, find
a. (ƒ ○ g) (-1).
ƒ(x) = 1/x , g(x) = 1/√ x + 2
문제 39d
Composition of Functions
In Exercises 39 and 40, find
d. (g ○ g) (x).
ƒ(x) = 1/x , g(x) = 1/√ x + 2
문제 41
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
문제 46
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
문제 47
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|
문제 49
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|
문제 51
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|
문제 53e
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
문제 53f
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
문제 54
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
cos 2θ + cos θ = 0
문제 54a
Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).
a. 𝔂 = ƒ(x - 5)
문제 54d
Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).
d. 𝔂 = ƒ(2x + 1)
문제 54e
Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).
e. 𝔂 = ƒ( x ) - 4
3
문제 55
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)
문제 56
Derive a formula for tan (A − B).
문제 58a
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).
문제 61
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.
문제 62
In Exercises 59–62, sketch the graph of the given function. What is the period of the function?
𝔂 = cos πx/2
