Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
Do each of the following equations represent a function?
![**Determining if Equations Represent a Function**
The following problem explores whether each given equation represents a function.
a. \(\frac{2}{3}x + y = 10\)
b. \(y^2 = x - 5\)
### Explanation:
- **Equation a:** To determine if \(\frac{2}{3}x + y = 10\) is a function, solve for \(y\):
\[
y = 10 - \frac{2}{3}x
\]
This is a linear equation in the form \(y = mx + b\), which represents a function because, for each input \(x\), there is exactly one output \(y\).
- **Equation b:** For \(y^2 = x - 5\), solving for \(y\) gives:
\[
y = \pm \sqrt{x - 5}
\]
This equation does not represent a function because, for some values of \(x\), there can be two different \(y\) values (positive and negative square roots).
### Graphical Representation:
- **Graph of Equation a:** A straight line, confirming a function relationship (one \(y\) for every \(x\)).
- **Graph of Equation b:** A parabola opening sideways, not representing a function because it fails the vertical line test (a vertical line can intersect the graph at more than one point).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5df5842e-eea8-4ec3-8ea8-c27cad8d62e1%2F3aa74009-b1b3-4766-a99e-03773b5f276f%2Fc7fe1w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determining if Equations Represent a Function**
The following problem explores whether each given equation represents a function.
a. \(\frac{2}{3}x + y = 10\)
b. \(y^2 = x - 5\)
### Explanation:
- **Equation a:** To determine if \(\frac{2}{3}x + y = 10\) is a function, solve for \(y\):
\[
y = 10 - \frac{2}{3}x
\]
This is a linear equation in the form \(y = mx + b\), which represents a function because, for each input \(x\), there is exactly one output \(y\).
- **Equation b:** For \(y^2 = x - 5\), solving for \(y\) gives:
\[
y = \pm \sqrt{x - 5}
\]
This equation does not represent a function because, for some values of \(x\), there can be two different \(y\) values (positive and negative square roots).
### Graphical Representation:
- **Graph of Equation a:** A straight line, confirming a function relationship (one \(y\) for every \(x\)).
- **Graph of Equation b:** A parabola opening sideways, not representing a function because it fails the vertical line test (a vertical line can intersect the graph at more than one point).
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