In Exercises 39–42, express the given quantity in terms of sin x and cos x.

sin (2π − x)

[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.

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

f(x) = x⁻⁵

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = |x| - 2

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

−cos 2πx

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

h(t) = |t³|

Refer to the given figure. Write the radius r of the circle in terms of α and θ.

Theory and Examples

The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.

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

Graph the functions in Exercises 37–56.

y = |x − 2|

Solving Trigonometric Equations

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

sin² θ = cos² θ

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

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.

Functions

In Exercises 1–6, find the domain and range of each function.

f(x) = 1 + x²

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

Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.

cos (x − π/2) = sin x

Graph the functions in Exercises 37–56.

y = (x + 2)³/² + 1

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) = sin³ x.

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = e⁻ˣ²

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

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = cos(x - 3) + 1

What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?

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.

Functions

In Exercises 1–6, find the domain and range of each function.

G(t) = 2/(t² − 16)

Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.

sin (x − π/2) = −cos x

Evaluate sin (5π/12).

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x² + 1

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x - sin x

In Exercises 19–32, find the (a) domain and (b) range.

_____

𝔂 = -1 + ∛ 2 - x

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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