Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 4

Chapter 3, Problem 4

Determine whether each equation defines y as a function of x. 2x + y = 8

Verified step by step guidance

Rewrite the given equation in terms of y to isolate it. Start with the equation 2x + y = 8 and subtract 2x from both sides to get y = -2x + 8.

Recall the definition of a function: For y to be a function of x, each input value of x must correspond to exactly one output value of y.

Examine the rewritten equation y = -2x + 8. Notice that for any given value of x, there is exactly one corresponding value of y because the equation is linear.

Conclude that the equation y = -2x + 8 defines y as a function of x because it satisfies the condition of having one unique output for each input.

If needed, you can also verify this by graphing the equation y = -2x + 8, which will produce a straight line, further confirming that it represents a function.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we must check if for every x, there is a unique y. This means that no x-value can produce multiple y-values.

Vertical Line Test

The vertical line test is a visual way to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the relation is not a function. This test helps to quickly assess the uniqueness of y-values for given x-values.

Solving for y

To analyze whether an equation defines y as a function of x, we often rearrange the equation to solve for y. In the case of 2x + y = 8, isolating y gives y = 8 - 2x, which clearly shows that for each x, there is a corresponding unique y, confirming that it is a function.

Recommended video:

5:02

Solving Logarithmic Equations

Related Practice

Textbook Question

Find the domain of each function. g(x) = 2/(x+5)

Textbook Question

Find the domain of each function. f(x) = x² - 2x - 15

views

Textbook Question

Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (4, -1) and (-6, 3)

views

Textbook Question

Use the graph of y = f(x) to graph each function g.

g(x) = f(x+1)

views

Textbook Question

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 4x + 9 and g(x) = (x-9)/4

views

Textbook Question

Determine whether each relation is a function. Give the domain and range for each relation. {(3, 4), (3, 5), (4, 4), (4, 5)}