/*! 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 55 Explain why the graph of the equ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 55

Explain why the graph of the equation is not the graph of a function. $$x=y^{2}$$

Short Answer

Expert verified
The graph of \( x = y^2 \) is not a function because each \( x \, (>0) \) value has two potential \( y \) values, failing the vertical line test.

Step by step solution

01

Understanding the Equation

The equation given is \( x = y^2 \). Notice how the variable \( x \) is expressed in terms of \( y \) squared.
02

Definition of a Function

A function assigns exactly one output \( y \) for every input \( x \). This means that for each \( x \) value, there should be only one corresponding \( y \) value.
03

Testing the Equation

Rewrite the equation as \( y = \pm \sqrt{x} \). This means for each positive \( x \), there are two \( y \) values (\( +\sqrt{x} \) and \( -\sqrt{x} \)).
04

Analyzing the Graph

The graph of \( x = y^2 \) is a sideways parabola. For any \( x > 0 \), there are two points on this graph corresponding to the same value of \( x \) (one in the upper half, \( y = +\sqrt{x} \), and one in the lower half, \( y = -\sqrt{x} \)).
05

Conclusion about Function

Since there can be more than one \( y \) value for a given \( x \) value, the graph does not represent a function.

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.

Parabola
A parabola is a symmetrical, curved shape that you often see in graphs. In usual math classes, we mostly deal with vertical parabolas, like the graph of the equation \( y = x^2 \). This kind of parabola opens upwards or downwards.
  • In a vertical parabola, each \( x \) value corresponds to exactly one \( y \) value.
  • This is a typical feature of functions, as they provide one output for each input.
The equation \( x = y^2 \) gives us a different type of parabola, which is not vertical but sideways. This sideways parabola does not meet the typical definition of a function because a single \( x \) value can correspond to multiple \( y \) values.
Function Definition
Understanding the definition of a function is crucial in mathematics. A function is a relation where every input (commonly \( x \)) has exactly one output (commonly \( y \)). This concept can be visualized as a machine that accepts an input and produces one consistent product.
  • If you can find more than one \( y \) value for a given \( x \), then it鈥檚 not a function.
  • Think of functions like vending machines: if you press a button (input) and get several products (outputs), the machine isn't functioning properly!
Multiple Y-Values
The equation \( x = y^2 \) can be rewritten as \( y = \pm \sqrt{x} \), indicating two potential outputs for each positive \( x \) value. This results in a single \( x \) associated with two \( y \) values, \(+\sqrt{x} \) and \(-\sqrt{x} \).
  • For instance, if \( x = 4 \), then \( y \) could be 2 or -2.
  • This situation contradicts the fundamental requirement of a function, where an \( x \) should map to exactly one \( y \).
Because of the possible multiple \( y \)-values, the graph of \( x = y^2 \) is not a graph of a function.
Horizontal Line Test
The horizontal line test is a visual way to determine if a graph represents a function.
  • Draw horizontal lines across the graph.
  • If any horizontal line crosses the graph at more than one point, the graph does not represent a function.
In the case of a sideways parabola like \( x = y^2 \), a horizontal line intersects the graph at two points for almost every \( x \).
This confirms that the graph fails the horizontal line test, indicating it's not a 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

Sketch the graph of the equation. $$y=|\sqrt{x}-1|$$

Find two positive real numbers whose sum is 40 and whose product is a maximum.

Estimate the solutions of the inequality. $$\left|1.2 x^{2}-10.8\right|>1.36 x+4.08$$

A hot-air balloon rises vertically from ground level as a rope attached to the base of the balloon is released at the rate of \(5 \mathrm{ft} / \mathrm{sec}\) (see the figure). The pulley that releases the rope is 20 feet from a platform where passengers board the balloon. Express the altitude \(h\) of the balloon as a function of time \(t\) (IMAGES CANNOT COPY).

An electric company charges its customers S0.0577 per kilowatt-hour (kwh) for the first 1000 kWh used, 0.0532 dollar for the next \(4000 \mathrm{kWh},\) and 0.0511 dollar for any kWh over 5000 . Find a piecewise-defined function \(C\) for a customer's bill of \(x\) kWh.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

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.