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Problem 1.3.65
Refer to the given figure. Write the radius r of the circle in terms of α and θ.
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 1.2.80
Can a function be both even and odd? Give reasons for your answer.
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 1.3.44
Evaluate cos (11π/12) as cos (π/4 + 2π/3).
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 1.3.38
What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?
Problem 1.3.36
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
sin (A − B) = sin A cos B − cos A sin B
Problem 1.1.18
Functions and Graphs
Find the natural domain and graph the functions in Exercises 15–20.
g(x) = √−x
Problem 1.2.43
Graph the functions in Exercises 37–56.
y = (x + 1)²/³
Problem 1.4.20
Finding a Viewing Window
In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.
f(x) = (x² − 1)/(x² + 1)
Problem 1.12
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = sec x tan x
Problem 1.37.1
In Exercises 37 and 38, write a piecewise formula for the function.
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Problem 1.23
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 2e⁻ˣ - 3
Problem 1.3.31
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
cos (x − π/2) = sin x
Problem 1.2.26
Shifting Graphs
The accompanying figure shows the graph of y = −x² shifted to four new positions. Write an equation for each new graph.
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Problem 1.1.21
Functions and Graphs
Find the domain of y = (x + 3) / (4 − √(x² − 9)).
Problem 1.2.4
Algebraic Combinations
In Exercises 3 and 4, find the domains of f, g, f/g and g/f.
f(x) = 1, g(x) = 1 + √x
Problem 1.2.21
Composition of Functions
A balloon’s volume V is given by V = s² + 2s + 3 cm³, where s is the ambient temperature in °C. The ambient temperature s at time t minutes is given by s = 2t − 3 °C. Write the balloon’s volume V as a function of time t.
Problem 1.1.14
Finding Formulas for Functions
Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.
Problem 1.20
In Exercises 19–32, find the (a) domain and (b) range.
____
𝔂 = -2 + √1 - x
Problem 1.1.10
Finding Formulas for Functions
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
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 1.35
Piecewise-Defined Functions
In Exercises 35 and 36, find the (a) domain and (b) range.
𝔂 = { √ -x, -4 ≤ x ≤ 0
{ √ x, 0 < x ≤ 4
Problem 1.1.12
Finding Formulas for Functions
A point P in the first quadrant lies on the graph of the function f(x) = √x. Express the coordinates of P as functions of the slope of the line joining P to the origin.
Problem 1.4
A hot-air balloon rising straight up from a level field is tracked by a range finder located 500 ft from the point of liftoff. Express the balloon’s height as a function of the angle the line from the range finder to the balloon makes with the ground.
Problem 1.4.26
Finding a Viewing Window
In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.
y = 3 cos 60x
Problem 1.4.33
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 four periods of the function f (x) = −tan 2x.
Problem 1.3.48
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
cos² 5π/12
