Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

2. Graphs of Equations

Two-Variable Equations

College Algebra

2. Graphs of Equations

Two-Variable Equations

Previous problemNext problem

1:54 minutes

Problem 11

Textbook Question

In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16

Verified step by step guidance

Rewrite the equation in terms of \( y \). Start with the given equation: \( x + y = 16 \).

Subtract \( x \) from both sides to isolate \( y \): \( y = 16 - x \).

Recognize that this equation is in the form \( y = f(x) \), where \( f(x) = 16 - x \).

Understand that for each value of \( x \), there is exactly one corresponding value of \( y \).

Conclude that the equation \( y = 16 - x \) defines \( y \) as a function of \( x \) because it passes the vertical line test.

Verified video answer for a similar problem:

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

Video duration:

Play a video:

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 between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, no two ordered pairs can have the same first element with different second elements. This concept is crucial for determining if an equation defines y as a function of x.

Vertical Line Test

The vertical line test is a visual way to determine if a curve is a graph of 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 whether y can be expressed uniquely for each x in a given equation.

Solving for y

To determine if an equation defines y as a function of x, it is often useful to solve the equation for y. If the equation can be rearranged to express y solely in terms of x, it indicates that y is a function of x. For example, in the equation x + y = 16, isolating y gives y = 16 - x, confirming that y is indeed a function of x.