/*! 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 58 Graph each function and state th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 58

Graph each function and state the domain and range. $$f(x)=-|x|+2$$

Short Answer

Expert verified
The graph of the function \(f(x)= -|x|+2\) is an inverted V shape that opens downward with its vertex at (0, 2). The domain of the function is all real numbers while the range is (-鈭�, 2].

Step by step solution

01

Understand the Function

The function \(f(x)= -|x|+2\) is an absolute value function, which typically has a shape of a 'V' or inverted 'V' and opens up if the coefficient of |x| is positive and opens down if the coefficient is negative. In this case, the function has a coefficient of -1, so the graph opens down. The +2 in the function represents a vertical shift where the vertex of the function moves up by 2 places.
02

Plot the Function

Draw the inverted 'V' shape of the absolute value function, reflecting the fact that it opens down due to the negative coefficient. The vertex of the function is at (0,2). For each positive integral value of x, plot a point at a corresponding y-value of \(2 - x\). Reflect these points across the y-axis to get the graph for negative values of x.
03

Determine the Domain and Range

The domain of a function is the set of all possible x-values, and in this case, x can be any real number which makes the domain all real numbers. The function will go no higher than the vertex, because it opens down, so the range is all numbers up to and including 2. So the range is (-鈭�, 2].

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 Domain and Range
The domain and range are fundamental concepts when working with functions, including absolute value functions.
  • The domain of a function refers to all the possible x-values (inputs) for which the function is defined. For the function \(f(x) = -|x| + 2\), there are no restrictions on x, meaning it can take any real number value. Thus, the domain is "all real numbers," often represented as \((-\infty, \infty)\).
  • The range of a function denotes all possible y-values (outputs) resulting from the domain. For our function, because it is an inverted 'V' opening downwards, the vertex is the highest point on the graph. As the graph decreases indefinitely, the range starts at \(-\infty\) and stretches up to the vertex y-value. Hence, the range is \((-\infty, 2]\).
Understanding the domain and range helps in sketching the graph and in solving real-world problems where certain outcomes are restricted.
Graphing Absolute Value Functions
Graphing functions can provide a visual understanding of how they behave. For absolute value functions, it's key to identify the shape and the direction it opens. Let's break it down for \(f(x) = -|x| + 2\):
  • The function represents an inverted 'V' due to the negative sign in front of the absolute value. This means the arms of the 'V' point downwards.
  • The vertex of the function is at \((0, 2)\), indicating where the highest tip of the graph starts.
  • From the vertex, as you move away from the y-axis, the graph materialsizes symmetrically. If you increase x by 1 in the positive direction, you decrease y by 1 to plot the point \((1, 1)\), and the same logic applies going left.
Reflect these points across the y-axis to depict negative x-values, ensuring symmetry which is characteristic of absolute value functions. A clear plot will reveal how the function intersects the axes and allows you to visualize its behavior across different inputs.
The Vertex of an Absolute Value Function
The vertex of an absolute value function is particularly important because it represents a pivotal point on the graph. In \(f(x) = -|x| + 2\), the vertex is at \((0, 2)\). Let's explore what this means:
  • The vertex in absolute value functions is the point where the graph makes a sharp turn. For an inverted 'V', like in this function, the vertex is the maximum point.
  • Here, \(x = 0\) gives the highest y-value possible, which is 2, showing that the vertex is positioned at the coordinate \((0, 2)\).
  • This vertex confirms the vertical shift dictated by the constant term in the function. Specifically, the "+2" means this maximum point is 2 units above where it would typically be at \((0, 0)\) in \(y = -|x|\).
Understanding the vertex's role helps depict the graph's orientation and allows accurate diagramming, aiding in identifying unique points such as intersections and axes behavior.

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

Set up a variation equation and solve for the requested value. Assume that the value of a machine varies inversely with its age. If a drill press is worth \(300\)dollar when it is 2 years old, find its value when it is 6 years old. How much has the machine depreciated in those 4 years?

Express each sentence as a formula. The force of gravity \(F\) varies directly with the product of the masses \(m_{1}\) and \(m_{2}\), and inversely with the square of the distance between them.

Explain the process used to find the \(x\)- and \(y\)-intercepts of the graph of a line.

Graph each equation using any method. $$y+x=-2$$

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$P(2,-7) \text { and } Q(-3,12)$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

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.