/*! 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 11 Determine whether \(y\) is a fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 11

Determine whether \(y\) is a function of \(x\) in each of the following equations. If the equation does not define a function, find a value of \(x\) that is associated with two different \(y\) values. a. \(y=x^{2}+1\) c. \(y=5\) b. \(y=3 x-2\) d. \(y^{2}=x\)

Short Answer

Expert verified
a, b, and c define functions. d does not; for \(x = 4\), \(y = 2\) and \(y = -2\).

Step by step solution

01

Analyze Equation a

Consider the equation \(y = x^2 + 1\). For each value of \(x\), there is a unique value of \(y\). Thus, \(y\) is a function of \(x\).
02

Analyze Equation b

Consider the equation \(y = 3x - 2\). For each value of \(x\), there is a unique value of \(y\). Thus, \(y\) is a function of \(x\).
03

Analyze Equation c

Consider the equation \(y = 5\). For each value of \(x\), there is a unique value of \(y\) (which is always 5). Thus, \(y\) is a function of \(x\).
04

Analyze Equation d

Consider the equation \(y^2 = x\). For a given \(x = 4\), \(y = 2\) and \(y = -2\). Therefore, \(x = 4\) is associated with two different \(y\) values, so \(y\) is not a function of \(x\) in this case.

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.

function analysis
Understanding the concept of functions is crucial in algebra. A function is a relationship where each input (or value of \(x\)) is related to exactly one output (or value of \(y\)). For example, in the equation \(y = x^2 + 1\), no matter what value you choose for \(x\), there is only one corresponding \(y\). This makes \(y\) a function of \(x\). Similarly, in the equations \(y = 3x - 2\) and \(y = 5\), each \(x\) value maps to one unique \(y\) value. However, not all relationships between \(x\) and \(y\) are functions. When analyzing an equation, we must check if the output for each input is unique.
unique values
Determining if an equation is a function involves checking the uniqueness of \(y\) values for each \(x\). In the function \(y = 3x - 2\), the expression shows that for every \(x\), there is a single, predictable \(y\). This confirms the uniqueness of values. Another example is \(y = 5\), where no matter what \(x\) is, \(y\) will always be 5. This still satisfies the condition of each \(x\) mapping to one unique \(y\) value. On the contrary, if an equation maps one \(x\) value to multiple \(y\) values, it fails the test of a function. This is evident in the equation \(y^2 = x\), where solving for \(y\) given an \(x\) can provide two distinct values for \(y\).
non-functions
Not every equation defines a function. A key characteristic of non-functions is the presence of multiple \(y\) values for a single \(x\) value. Let's scrutinize the equation \(y^2 = x\). If \(x = 4\), solving \(y^2 = 4\) gives two solutions: \(y = 2\) and \(y = -2\). Accordingly, because one value of \(x\) leads to multiple \(y\) values, this relationship does not fulfill the definition of a function. Recognizing non-functions is essential for correctly interpreting and applying algebraic equations.
algebraic equations
Algebraic equations form the backbone of analyzing functions in algebra. They represent mathematical relationships and can be used to define functions or identify non-functions. Having a strong grasp of solving and interpreting these equations helps in identifying whether they meet the criteria for functions. As seen, the equations \(y = x^2 + 1\), \(y = 3x - 2\), and \(y = 5\) each uniquely map \(x\) to \(y\). In contrast, the equation \(y^2 = x\) illustrates a non-function scenario. Understanding and solving these equations is key to mastering algebraic concepts and analyzing function properties accurately.

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

Given here is a table of salaries taken from a survey of recent graduates (with bachelor degrees) from a well-known university in Pittsburgh. $$ \begin{array}{cc} \hline \begin{array}{c} \text { Salary } \\ \text { (in thousands) } \end{array} & \begin{array}{c} \text { Number of Graduates } \\ \text { Receiving Salary } \end{array} \\ \hline 21-25 & 2 \\ 26-30 & 3 \\ 31-35 & 10 \\ 36-40 & 20 \\ 41-45 & 9 \\ 46-50 & 1 \\ \hline \end{array} $$ a. How many graduates were surveyed? \(\mathbf{b}\). Is this quantitative or qualitative data? Explain. c. What is the relative frequency of people having a salary between \(\$ 26,000\) and \(\$ 30,000 ?\) d. Create a histogram of the data.

Determine the domain of each of the following functions. Explain your answers. \(f(x)=4 \quad g(x)=3 x+5 \quad h(x)=\frac{x-1}{x-2}\) \(F(x)=x^{2}-4 \quad G(x)=\frac{x-1}{x^{2}-4} \quad H(x)=\sqrt{x-2}\)

Choose a paragraph of text from any source and construct a histogram of word lengths (the number of letters in the word). If the same word appears more than once, count it as many times as it appears. You will have to make some reasonable decisions about what to do with numbers, abbreviations, and contractions. Compute the mean and median word lengths from your graph. Indicate how you would expect the graph to be different if you used: a. A children's book c. A medical textbook b. A work of literature

(Use of calculator or other technology recommended.) Use the following table to generate an estimate of the mean age of the U.S. population. Show your work. (Hint: Replace each age interval with an age approximately in the middle of the interval.) $$ \begin{aligned} &\text { Ages }\\\ &\text { of US. Population in } 2004\\\ &\begin{array}{lr} \hline \begin{array}{c} \text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} \text { Population } \\ \text { (thousands) } \end{array} \\ \hline \text { Under 10 } & 39,677 \\ 10-19 & 41,875 \\ 20-29 & 40,532 \\ 30-39 & 41,532 \\ 40-49 & 45,179 \\ 50-59 & 35,986 \\ 60-74 & 31,052 \\ 75-84 & 12,971 \\ 85 \text { and over } & 4,860 \\ \hline \text { Total } & 293,655 \\ \hline \end{array} \end{aligned} $$

Assume that for persons who earn less than \(\$ 20,000\) a year, income tax is \(16 \%\) of their income. a. Generate a formula that describes income tax in terms of income for people earning less than \(\$ 20,000\) a year. b. What are you treating as the independent variable? The dependent variable? c. Does your formula represent a function? Explain. d. If it is a function, what is the domain? The range?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.