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

𝔂 = 4 sin ( 1 )

x

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

𝔂 = 2 + 3x² .

x² + 4

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

________

𝔂 = 5 - √ x² - 2x - 3

Functions and Graphs

Graph the following equations and explain why they are not graphs of functions of x.

a. |x| + |y| = 1

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

a. <IMAGE>

Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

a. y = 2x − 3

Composition of Functions

Copy and complete the following table.

a. <IMAGE>

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

a. <IMAGE>

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.)

<IMAGE>

Functions and Graphs

Graph the following equations and explain why they are not graphs of functions of x.

a. |y| = x

The Greatest and Least Integer Functions

For what values of x is

b. ⌈x⌉ = 0

Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

b. <IMAGE>

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

b. <IMAGE>

Composition of Functions

Copy and complete the following table.

b. <IMAGE>

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.

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

b. <IMAGE>

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?

b. f/g

Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

b. <IMAGE>

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

b. <IMAGE>

Find the largest interval on which the given function is increasing.

b. ƒ(x) = (x + 1)⁴

Find the largest interval on which the given function is increasing.

c. g(x) = (3x - 1)¹/³

Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

c. y = x¹/⁴

The accompanying figure shows the graph of a function f(x) with domain [0,2] and range [0,1]. Find the domains and ranges of the following functions, and sketch their graphs.

<IMAGE>

c. 2f(x)

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

c. g(g(−1))

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?

d. f² = ff

Composition of Functions

Copy and complete the following table.

d. <IMAGE>

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

e. g(f(0))

Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

f. y = √(x³ − 3)

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

f. f(g(1/2))