0. Functions

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

The Greatest and Least Integer Functions

Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.

The Greatest and Least Integer Functions

For what values of x is

b. ⌈x⌉ = 0

Can a function be both even and odd? Give reasons for your answer.

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

Theory and Examples

The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.

a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)

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Theory and Examples

In Exercises 69 and 70, match each equation with its graph. Do not use a graphing device, and give reasons for your answer.

a. y = x⁴

b. y = x⁷

c. y = x¹⁰

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Industrial costs A power plant sits next to a river where the river is 800 ft wide. Laying a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs $180 per foot across the river and $100 per foot along the land.

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b. Generate a table of values to determine whether the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

[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.

[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.

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)

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 upper branch of the hyperbola y² − 16x² = 1.