/*! 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 15 Draw the graph of a function \(f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 15

Draw the graph of a function \(f\) that is 1 -to- 1 and a function \(g\) that is not 1 -to-1.

Short Answer

Expert verified
Function \(f(x) = x\) is an example of a one-to-one function. Its graph is a straight line passing through the origin, and it passes the horizontal line test. Function \(g(x) = x^2\), on the other hand, is a function that is not one-to-one. Its graph is a parabola and fails the horizontal line test.

Step by step solution

01

Draw Function f

To begin with, draw a function \(f\) which is 1 -to- 1. An example of such a function could be \(f(x) = x\). To represent this graphically, plot a straight diagonal line passing through the origin on the Cartesian plane.
02

Check the One-to-One Property of Function f

One way to check if a function is one-to-one is by applying the horizontal line test. If any horizontal line passes through the graph of the function more than once, then the function is not one-to-one. For function f, as it is a straight line, any horizontal line will only touch the graph at a single point, therefore it passes the horizontal line test and is one-to-one.
03

Draw Function g

Next, a function \(g\) that is not 1 -to-1 is required. An example can be \(g(x) = x^2\). To illustrate this visually, plot a parabola on the Cartesian plane.
04

Check the One-to-One Property of Function g

Again, apply the horizontal line test to function \(g\). For \(g(x) = x^2\), a horizontal line will intersect the graph at more than one point. Therefore, it fails the horizontal line test, proving that function g is not one-to-one.

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.

Horizontal Line Test
The horizontal line test is a useful tool to determine if a function is one-to-one. When a function is one-to-one, each output value is linked to only one input value. To perform the test:
  • Imagine drawing various horizontal lines (parallel to the x-axis) across the graph of the function.
  • Observe if any line intersects the graph at more than one point.
If a horizontal line only crosses the graph at a single point each time, the function is one-to-one. However, if any horizontal line crosses the graph more than once, the function is not one-to-one. For example, the graph of the function \(f(x) = x\) will only be touched once per horizontal line, confirming it's one-to-one. In contrast, the function \(g(x) = x^2\) would be intersected twice by many horizontal lines, demonstrating it's not one-to-one.
Function Graphs
Understanding function graphs is crucial when analyzing different mathematical functions. A function graph visually represents the relationship between inputs (x-values) and outputs (y-values). By plotting points on a graph:
  • We can observe patterns or shapes that help identify the type of function.
  • These visualizations can provide insights into the nature of the function, such as symmetry, intercepts, and behavior at infinity.
For instance, linear functions, like \(f(x) = x\), are depicted as straight lines. Meanwhile, quadratic functions, like \(g(x) = x^2\), form parabolas that open upwards. By analyzing these graphs, we can apply tests, like the horizontal line test, to determine properties such as the function鈥檚 one-to-one nature.
Parabolas
Parabolas are distinctive U-shaped graphs often associated with quadratic functions. The standard form of a quadratic function is \(y = ax^2 + bx + c\). In the context of one-to-one functions:
  • Parabolas are crucial because they often fail the horizontal line test.
  • They are symmetrical, meaning that for most horizontal lines, there will be two points of intersection for the same y-value.
This symmetry showcases why parabolas like the graph of \(g(x) = x^2\) are not one-to-one. However, restricting the domain (considering only one side of the parabola) can turn such a function into a one-to-one mapping by focusing on one unique portion of the graph, though this breaks its natural form.
Linear Functions
Linear functions are one of the simplest types of functions, represented graphically as straight lines. Characterized by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept:
  • They have a constant rate of change and are often easy to analyze.
  • For any linear function, a horizontal line will intersect at most one point, ensuring it is always one-to-one.
An example of a one-to-one linear function is \(f(x) = x\), which passes through the origin with a slope of one. Unlike parabolas, linear graphs possess no symmetry that would result in double intersections with any horizontal line, solidifying their one-to-one status automatically.

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

For Problems 7 through 9 determine whether the relationship described is a function. If the relationship is a function, (a) what is the domain? the range? (b) is the function 1 -to- 1 ? $$ \begin{array}{ll} \text { Input } & \text { Output } \\ \hline \sqrt{2} & 0 \\ 2 \sqrt{2} & 0 \\ 3 \sqrt{2} & 0 \\ 4 \sqrt{2} & 0 \end{array} $$

Filene's Basement regularly marks down its merchandise. A discounted item now costs \(D\) dollars. This is after a \(p\) -percent markdown. Express the initial price of the item in terms of \(p\) and \(D\). Try out your answer in the concrete case of an item that now costs \(\$ 100\) after a 20-percent markdown. Why should your answer not be \(\$ 120\) ?

Two bears, Bruno and Lollipop, discover a patch of huckleberries one morning. The patch covers an area of \(A\) acres and there are \(X\) bushels of huckleberries per acre. Bruno eats \(B\) bushels of huckleberries per hour; Lollipop can devour \(L\) bushels of huckleberries in \(C\) hours. Express your answers to parts (a) and (b) in terms of any or all of the constants \(A, X, B, L\), and \(C .\) (a) Express the number of bushels of huckleberries the two bears eat as a function of \(t\), the number of hours they have been eating. (b) In \(t\) hours, how many acres of huckleberries can the two bears together finish off? (c) Assuming that after \(T\) hours the bears have not yet finished the berry patch, how many hours longer does it take them to finish all the huckleberries in the patch? Express your answers in terms of any or all of the constants \(A, X, B, L, C\), and \(T\). If you are having difficulty, use this time-tested technique: Give the quantity you are looking for a name. (Avoid the letters already standing for something else.)

Express the area of a circle as a function of: (a) its diameter, \(d\). (b) its circumference, \(c .\)

For Problems 33 through 35, if the interval is written using inequalities, write it using interval notation; if it is expressed in interval notation, rewrite it using inequalities. In all cases, indicate the interval on the number line. $$ \text { (a) }-1 \leq x \leq 3 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.