Textbook Question
Use the vertical line test to identify graphs in which y is a function of x.
Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 11
In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16
Verified step by step guidance1
Start by understanding the problem: we need to determine if the equation \(x + y = 16\) defines \(y\) as a function of \(x\). A function means for each \(x\) there is exactly one \(y\) value.
Rewrite the equation to solve for \(y\) in terms of \(x\). Subtract \(x\) from both sides to isolate \(y\): \(y = 16 - x\).
Observe the expression \(y = 16 - x\). For every value of \(x\), there is exactly one corresponding value of \(y\) calculated by subtracting \(x\) from 16.
Since each \(x\) corresponds to exactly one \(y\), the equation defines \(y\) as a function of \(x\).
Conclude that the equation \(x + y = 16\) does define \(y\) as a function of \(x\) because it can be expressed as \(y = 16 - x\), which passes the vertical line test.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of a Function
A function is a relation where each input (x-value) corresponds to exactly one output (y-value). This means for every x, there is only one y. Understanding this helps determine if an equation defines y as a function of x.
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Solving for y in Terms of x
To check if y is a function of x, solve the equation for y explicitly. If y can be expressed as a single formula involving x without ambiguity, then y is a function of x.
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Vertical Line Test
The vertical line test is a graphical method to determine if y is a function of x. If any vertical line intersects the graph of the equation more than once, y is not a function of x.
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Related Practice
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