/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 In \(13-20\) : a. Graph each fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 17

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(y=\frac{1}{2} x\)

Short Answer

Expert verified
The function is a direct variation and is one-to-one.

Step by step solution

01

Identify the Function Type

The function given is \( y = \frac{1}{2}x \). This is a linear function in the form \( y = mx + b \) where \( m = \frac{1}{2} \) and \( b = 0 \). The graph of this function will be a straight line passing through the origin (0,0).
02

Graph the Function

To graph \( y = \frac{1}{2}x \), plot the y-intercept (0,0). Next, use the slope \( \frac{1}{2} \) to find another point. Starting at (0,0), move up 1 unit and right 2 units to reach (2,1). Draw a line through these points, extending in both directions. This line represents the function.
03

Determine if the Function is a Direct Variation

Direct variation is a relationship of the form \( y = kx \), where \( k eq 0 \). The given function \( y = \frac{1}{2}x \) matches this form with \( k = \frac{1}{2} \). Hence, the function is a direct variation.
04

Check if the Function is One-to-One

A function is one-to-one if each output (y-value) is paired with exactly one input (x-value). Linear functions of the form \( y = mx \) are one-to-one as long as \( m eq 0 \). Since \( m = \frac{1}{2} \), the function is one-to-one, as it passes the horizontal line test (each horizontal line intersects the graph at most once).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Direct Variation
The concept of direct variation is quite straightforward. In mathematics, a direct variation relationship can be described by the equation \( y = kx \), where \( k \) is a non-zero constant.
This means that as one variable changes, the other changes at a constant rate.
  • For instance, in the function \( y = \frac{1}{2}x \), \( k \) is \( \frac{1}{2} \).
  • The line of the graph will always pass through the origin (0,0), which is a hallmark of direct variation.
  • These types of functions are useful in many real-life scenarios where one quantity depends directly on another, such as speed and distance.
So, whenever you see a function of the form \( y = kx \) without a constant being added or subtracted, it's a perfect example of direct variation.
One-to-One Functions
Understanding one-to-one functions is key in determining the uniqueness of outputs. A function is one-to-one if every output corresponds with exactly one input.
In other words, no two different x-values have the same y-value.
This is important because one-to-one functions have the property that they can be inverted.
  • For linear functions with the form \( y = mx + b \), if \( m eq 0 \), the function is one-to-one.
  • The graph must pass the horizontal line test, meaning any horizontal line drawn through the graph hits no more than one point.
In the case of the function \( y = \frac{1}{2}x \), since the slope \( m \) is \( \frac{1}{2} \), which is not zero, this function is indeed one-to-one.
Graphing Linear Equations
Graphing linear equations is an essential skill whether you're just starting with algebra or exploring advanced mathematics. These equations create straight lines when graphed on a coordinate plane.
The form \( y = mx + b \) is called the slope-intercept form, where \( m \) is the slope, and \( b \) is the y-intercept.
  • The slope \( m \) tells us how steep the line is. In \( y = \frac{1}{2}x \), the slope is \( \frac{1}{2} \), indicating an upward incline that rises 1 unit for every 2 units it moves right.
  • The y-intercept \( b = 0 \) in this equation tells us the line passes through the origin (0,0).
  • To graph this equation, you start at the y-intercept and use the slope to determine the direction and slant of the line.
  • For example, from (0,0), you would move up 1 unit and over 2 units to plot the next point.
By connecting these points with a straight line, you can visualize the behavior and characteristics of the linear function.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \\{(1,4),(2,7),(1,10),(4,13)\\} $$

\(\operatorname{In} 19-22,\) let \(\mathrm{f}(x)=|x| \cdot\) Find \(\mathrm{f}(\mathrm{g}(x))\) and \(\mathrm{g}(\mathrm{f}(x))\) for each given function. $$ g(x)=2 x+3 $$

a. Sketch the graph of \(y=x^{2}\) b. Sketch the graph of \(y=-x^{2}\) c. Describe the graph of \(y=-x^{2}\) in terms of the graph of \(y=x^{2}\) . d. What transformation maps \(y=x^{2}\) to \(y=-x^{2} ?\)

In \(12-17,\) use a graph to find the solution set of each inequality. $$ x^{2}-2 x+1 < 0 $$

In \(6-12,\) tell whether the variables vary directly, inversely, or neither. Each day, Jaymee works for 6 hours typing \(p\) pages of a report at a rate of \(m\) minutes per page.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.