/*! 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 9 A function \(f\) is given. (a) U... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 9

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$f(x)=x-1$$

Short Answer

Expert verified
The domain and range of \(f(x) = x - 1\) are both \((-\infty, \infty)\).

Step by step solution

01

Understanding the Function

The given function is a linear function, specifically, \(f(x) = x - 1\). A linear function of the form \(f(x) = mx + c\) is represented by a straight line with a slope \(m\) and a y-intercept \(c\). In this function, the slope \(m\) is 1 and the y-intercept \(c\) is -1.
02

Using a Graphing Calculator

To draw the graph of the function \(f(x) = x - 1\), enter it into a graphing calculator. The graph will be a straight line that crosses the y-axis at -1 and has a slope of 1, meaning it rises one unit for every one unit it moves to the right.
03

Finding the Domain from the Graph

The domain of a function is the set of all possible input values (x-values). For a linear function, such as \(f(x) = x - 1\), there are no restrictions on the x values. Therefore, the domain of this function is all real numbers, denoted as \((-\infty, \infty)\).
04

Finding the Range from the Graph

The range of a function is the set of all possible output values (y-values). Similar to the domain, for a linear function like \(f(x) = x - 1\), the function can produce all real numbers as output. Thus, the range is all real numbers, denoted as \((-\infty, \infty)\).

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.

Graphing Calculator
A graphing calculator is an invaluable tool when it comes to visualizing mathematical functions. It enables you to plot functions like a linear function, which is simply a straight line described by an equation such as \(f(x) = x - 1\). Using a graphing calculator to plot this function involves inputting the equation into the calculator.

When you input \(f(x) = x - 1\), the calculator draws a straight line extending infinitely in both directions. The line鈥檚 slope of 1 means that for every step to the right on the x-axis, the line rises one unit on the y-axis. The y-intercept, which is -1 in this case, indicates the point where the line crosses the y-axis.

Plotting the function using a graphing calculator provides a clear visual representation that helps identify key features of the function, such as the domain and range.
Domain of a Function
The domain of a function refers to all the possible input values, or x-values, that you can use with the function. For linear functions like \(f(x) = x - 1\), there are no restrictions on x-values.

This means the function can handle any real number as an input. Therefore, for \(f(x) = x - 1\), the domain is all real numbers, represented in mathematical notation as
  • \((-\infty, \infty)\).
Visualizing this on a graph, you would see the line extending endlessly in both left and right directions along the x-axis. This unbounded extent confirms the function's comprehensive domain, allowing for exploration of input across the entire number line.
Range of a Function
The range of a function is the set of all possible output values, or y-values, it can produce. For our linear function \(f(x) = x - 1\), every real number can be an output. The graph demonstrates that as x-values increase or decrease without limit, so do the y-values.

Just as the domain is limitless, the range in this case also encompasses all real numbers. In mathematical terms, the range is also expressed as:
  • \((-\infty, \infty)\).
This infinite vertical extension on both sides of the graph illustrates that any real value can be a result of the function. The line鈥檚 continual rise and fall confirm that \(f(x) = x - 1\) isn鈥檛 bound by ceilings or floors, providing vast possibilities for y-values.

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

A function is given. (a) Find all the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. (b) Find the intervals on which the function is increasing and on which the function is decreasing. State each answer correct to two decimal places. $$f(x)=3+x+x^{2}-x^{3}$$

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. To evaluate \(f(x)\), divide the input by 3 and add \(\frac{2}{3}\) to the result.

Evaluate the function at the indicated values. $$\begin{array}{l} f(x)=x^{3}-4 x^{2} ; \\ f(0), f(1), f(-1), f\left(\frac{3}{2}\right), f\left(\frac{x}{2}\right), f\left(x^{2}\right) \end{array}$$

Find the domain of the function. $$f(x)=\sqrt{x-5}$$

Evaluate the function at the indicated values. $$\begin{aligned} &h(t)=t+\frac{1}{t} ; \\ &h(1), h(-1), h(2), h\left(\frac{1}{2}\right), h(x), h\left(\frac{1}{x}\right) \end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Pure Maths

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.