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Problem 1.3.40
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
sin (2π − x)
Problem 1.1.72
[Technology Exercise]
a. Graph the functions f(x) = 3/(x − 1) and g(x) = 2/(x + 1) together to identify the values of x for which
3/(x − 1) < 2/(x + 1)
b. Confirm your findings in part (a) algebraically.
Problem 1.2.80
Can a function be both even and odd? Give reasons for your answer.
Problem 1.19
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = |x| - 2
Problem 1.3.18
Graph the functions in Exercises 13–22. What is the period of each function?
−cos 2πx
Problem 1.3.44
Evaluate cos (11π/12) as cos (π/4 + 2π/3).
Problem 1.3.50
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
sin² 3π/8
Problem 1.1.65
Theory and Examples
The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.
Problem 1.1.52
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
g(x) = x⁴ + 3x² − 1
Problem 1.2.39
Graph the functions in Exercises 37–56.
y = |x − 2|
Problem 1.3.54
Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
Problem 1.2.76
Graphing
In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.
y = (−2x)²/³
Problem 1.1.75
A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs $5/ft for the legs and $10/ft for the hypotenuse, write the total cost C of construction as a function of h.
Problem 1.8
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = e⁻ˣ²
Problem 1.3.1
Radians and Degrees
On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4π/5 radians? (b) 110°?
Problem 1.1.54
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
g(x) = x/(x² − 1)
Problem 1.13
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x⁴ + 1
x³ - 2x
Problem 1.1.50
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = x² + x
Problem 1.1.37
Increasing and Decreasing Functions
Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
y = −x³
Problem 1.1.1
Functions
In Exercises 1–6, find the domain and range of each function.
f(x) = 1 + x²
Problem 1.27
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = cos(x - 3) + 1
Problem 1.4.35
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph the function f (x) = sin 2x + cos 3x.
Problem 1.1.35
The Greatest and Least Integer Functions
Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.
Problem 1.1.6
Functions
In Exercises 1–6, find the domain and range of each function.
G(t) = 2/(t² − 16)
Problem 1.3.34
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
sin (x − π/2) = −cos x
Problem 1.1.3
Functions
In Exercises 1–6, find the domain and range of each function.
F(x) = √(5x + 10)
Problem 1.2.48
Graph the functions in Exercises 37–56.
y = (x + 2)³/² + 1
Problem 1.14
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x - sin x
Problem 1.28
In Exercises 19–32, find the (a) domain and (b) range.
_____
𝔂 = -1 + ∛ 2 - x
Problem 1.1.71
[Technology Exercise]
a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which
x/2 > 1 + 4/x
b. Confirm your findings in part (a) algebraically.