/*! 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 25 Graph the function by hand. $$... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 25

Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} 1, & x \leq 2 \\ x, & 2

Short Answer

Expert verified
The resulting graph is a horizontal line passing through (2,1) for \(x \leq 2\), followed by a straight line segment starting from (2,2) with a 45-degree slope until \(x = 5\). The connecting point at \(x = 2\) is a closed circle for the first segment and an open circle for the second segment.

Step by step solution

01

Understand the First Piece of the function

For \(x \leq 2\), the function gives an output of 1. This piece can be plotted directly on the graph as a straight line parallel to the x-axis and going through the point (2,1). Note that the line should include the point where \(x = 2\).
02

Understand the Second Piece of the function

For the range \(2 < x \leq 5\), the function equates to \(x\), which represents a straight line with a slope of 45 degrees. This means that as \(x\) increases by one unit, \(f(x)\) also increases by one unit. The line should start from the point where \(x = 2\) (but this point is not included in this piece of function) and end at the point where \(x = 5\).
03

Combine the pieces to create the full graph

After plotting both pieces, check that the pieces join correctly at \(x = 2\). Since the point \(x=2\) is included in the first piece but not in the second, there should be a hollow circle at the point where the second piece starts (namely at (2,2)). This indicates that this point is not included in the function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Graphing functions, especially piecewise functions, can be a bit tricky at first. But once you understand the steps, it's as easy as pie. A piecewise function provides different expressions for different intervals of the domain. In our example here, the function has two pieces. Both pieces are graphed for specific ranges of the variable \(x\).
  • The first piece is a constant value of 1 for all \(x\leq 2\). This results in a horizontal line. When graphing, ensure you mark this section with a solid dot at the endpoint to show that it includes \(x = 2\).
  • The second piece follows \(f(x) = x\) for \(2 < x \leq 5\). This results in a line with a slope of 1, starting from just after \(x = 2\) up to \(x = 5\).

The concept is to graph each piece over its designated interval and then combine them carefully, ensuring continuity where needed to reflect the function accurately.
Mathematical Notation
Mathematical notation is like a language that allows us to define functions clearly and concisely. With piecewise functions, understanding the notation is crucial. In the given function:
  • The curly brackets \(\{\ldots\}\) indicate the function is defined in pieces.
  • The inequality \(x \leq 2\) tells us when to use the constant 1 as the function's value.
  • The expression \(2 < x \leq 5\) following \(x\) not only shows that the function is linear but also specifies its valid range.

Remember, different parts of the function are tied to specific conditions on \(x\). Solid and hollow circles are sometimes used in graphs to show whether endpoints are included or excluded.
Step-by-Step Solutions
When solving math problems, step-by-step solutions are incredibly helpful in breaking down complex ideas into smaller, digestible pieces.
In the given exercise, we break down the graphing task into steps:
  • Understanding the first piece: Recognize that for \(x \leq 2\), the output is always one, creating a flat line.
  • Understanding the second piece: Realize that \(f(x)\) equals \(x\) itself for \(2 < x \leq 5\). This makes a diagonal line on the graph.
  • Combining the pieces: Understand the endpoint inclusions or exclusions to ensure accurate representation.

Breaking it down this way makes the process manageable and less daunting. This detailed methodology ensures a deep understanding of how to approach piecewise function problems.

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Most popular questions from this chapter

Solve the inequality algebraically and graphically. Express your anstoer in intereal notation. $$-x+6 \leq 2 x+9$$

Sketch by hand the graph of the line with slope \(\frac{5}{3}\) that passes through the point \((-2,6) .\) Find the equation of this line.

Solve the inequality. Express your answer in interval notation. $$x+4

Applications In this set of exercises you will use the concepts of intersection of lines and linear inequalities to study real-world problems. Cost and Revenue In Exercises \(69-72,\) for each set of cost and revenue functions, (a) find the break-even point and (b) calculate the values of \(q\) for cohich retucnue exceeds cost. $$\begin{aligned} &C(q)=10 q+200\\\ &R(q)=15 q \end{aligned}$$

This set of exercises will draw on the ideas presented in this section and your general math background. Show that \(|x-k|=|k-x|\), where \(k\) is any real number.
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