/*! 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 Graph the function and determine... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 55

Graph the function and determine the interval(s) for which \(f(x) \geq 0\) . \(f(x)=4-x\)

Short Answer

Expert verified
The interval for which \(f(x) \geq 0\) is \([- \infty, 4]\).

Step by step solution

01

Graph the function

The function \(f(x)=4-x\) is a linear function with slope -1 and y-intercept 4. Start by plotting the y-intercept (0,4). Since the slope is -1, you can move down 1 unit and to the right 1 unit to find the next point (1,3). Repeat this process to get a couple more points and then draw the line through those points.
02

Determine when the function is greater than or equal to 0

The function \(f(x)=4-x\) will be greater than or equal to 0 when \(4-x \geq 0\). Solving this inequality gives \(x \leq 4\). So, the interval for which \(f(x) \geq 0\) is \([- \infty, 4]\).
03

Confirm with the graph

Confirm the interval solution by referencing the graph. Here, the graph of \(f(x) = 4 - x\) is indeed nonnegative (above or on the x-axis) from x-values -infinity up to 4, including 4.

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.

Understanding Inequality Solution
Solving an inequality often involves finding the values that make the inequality true. In the case of a linear function like \( f(x) = 4 - x \), we want to determine where \( f(x) \) is greater than or equal to zero. To solve \( 4 - x \geq 0 \), follow these simple steps:
  • Manipulate the inequality to isolate \( x \) by adding \( x \) to both sides, yielding \( 4 \geq x \).
  • Rewriting it, we find \( x \leq 4 \). This means \( f(x) \) is non-negative for any \( x \) less than or equal to 4.
Remember, we are treating the inequality like an equation, but keeping in mind that our solution includes all values making the inequality true, forming an interval. This interval is \([-\infty, 4]\), representing all possible \( x \) values from negative infinity to 4, inclusively.
Graphing Linear Equations
Graphing a linear function like \( f(x) = 4 - x \) helps visually understand the relationship between \( x \) and \( f(x) \). Every linear function can be graphically represented as a straight line. Here鈥檚 how you can graph it step-by-step:
  • Identify the y-intercept, which is the constant term \( 4 \). Plot the point \((0,4)\) on the graph.
  • Identify the slope, which for this function is \(-1\). The slope tells us to go down 1 unit and 1 unit to the right from any point on the line. From \((0, 4)\), moving down 1 unit and right 1 unit, leads us to \((1, 3)\). These steps can be repeated to form a straight line along these points.
The plotted line represents the function across all \( x \)-values, illustrating where \( f(x) = 4 - x \) is positive, negative, or zero on the graph. By examining the line, you confirm visually where the function is greater than or equal to zero.
Exploring Slope and Y-Intercept
The slope and y-intercept are key components to understanding linear functions. The slope determines the angle or steepness of the line, while the y-intercept indicates where the line crosses the y-axis. For \( f(x) = 4 - x \):
  • The slope is \(-1\). This negative slope means the line decreases from left to right, embodying a downward trend.
  • The y-intercept is the point where the line crosses the y-axis, given by the value 4 in our function. Thus, the line crosses at \((0, 4)\).
Understanding how the slope and y-intercept impact the graph gives better insights into how changes in \( x \) affect \( f(x)\). For example, every increase in \( x \) decreases \( f(x) \) by 1, because of the \(-1\) slope, reflecting how the graph of \( f(x) = 4 - x \) behaves across its domain.

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 with a square root cannot have a domain that is the set of real numbers.

Think About lt Restrict the domain of \(f(x)=x^{2}+1\) to \(x \geq 0 .\) Use a graphing utility to graph the function. Does the restricted function have an inverse function? Explain.

Matching and Determining Constants In Exercises \(85-88\) , match the data with one of the following functions $$ f(x)=c x, g(x)=c x^{2}, h(x)=c \sqrt{|x|}, \text { and } r(x)=\frac{c}{x} $$ and determine the value of the constant \(c\) that will make the function fit the data in the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {-4} & {-1} & {0} & {1} & {4} \\\ \hline y & {-32} & {-2} & {0} & {-2} & {-32} \\ \hline\end{array} $$

True or False? In Exercises \(71-74\) , determine whether the statement is true or false. Justify your answer. Think About It You can use either of two methods to graph a function: plotting points or translating a parent function as shown in this section. Which method of graphing do you prefer to use for each function? Explain. (a) \(f(x)=3 x^{2}-4 x+1\) (b) \(f(x)=2(x-1)^{2}-6\)

Mathematical models that involve both direct and inverse variation are said to have _________ variation.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.