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Chapter 1: Problem 22

Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} 3, & x \leq-1 \\ -x+2, & x>-1 \end{array}\right.$$

Short Answer

Expert verified
The graph of the given piecewise function consists of a horizontal line at \(y = 3\) for \(x \leq -1\), and the line \(y = -x + 2\) for \(x > -1\).

Step by step solution

01

Understanding the Piecewise Definition

This function is defined differently on different intervals. For \(x \leq -1\), \(f(x) = 3\). This is a constant function that will plot as a horizontal line at \(y = 3\). For \(x > -1\), \(f(x) = -x+2\). This is a linear function having a slope of -1 and y-intercept of 2.
02

Graphing \(f(x) = 3\) on the interval \(x \leq -1\)

Start by plotting the constant function \(f(x) = 3\) for \(x \leq -1\). Draw a horizontal line at \(y = 3\) starting from \(x = -1\) and extend it to the left. Include a filled circle at \((-1, 3)\) to indicate that the value at \(x = -1\) is included in this piece.
03

Graphing \(f(x) = -x+2\) on the interval \(x > -1\)

Next, plot the linear function \(f(x) = -x+2\) for \(x > -1\). Use the y-intercept, which is \(2\), and the negative slope to draw the line. Start by placing an open circle at the point \((-1, 3)\) because the value at \(x = -1\) is not included in this part of the function. Then, since the slope is -1, for every 1 unit move to the right, move 1 unit down, and draw the line.

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Key Concepts

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

Graphing Piecewise Functions
Graphing piecewise functions involves understanding how to plot separate sections of a function that are defined by different rules over different intervals of the input variable, usually denoted as "x." In the given piecewise function:
  • For values of \(x \leq -1\), the function is constant, \(f(x) = 3\).
  • For values of \(x > -1\), the function is \(f(x) = -x + 2\).
This means that two different graphs are combined into one complete graph, but not overlapping in the same region. Here's how to work through the process of graphing it:
  • Identify intervals and corresponding functions for each region.
  • Plot each section individually, maintaining consistent conditions for boundaries like open or closed circles to show included or excluded values.
  • Visualize transitioning from one piece to the other smoothly, respecting these boundaries.
Constant Function
A constant function is a type of function where the output value does not change regardless of the input value. It is represented by a single horizontal line on a graph.For our function, on the interval \(x \leq -1\), the function value is always \(f(x) = 3\), creating a constant horizontal line at \(y = 3\). This reflects the idea that for any input within this range, the output is a fixed number.
The distinguishing characteristics of a constant function are:
  • It appears visually as a horizontal line in its defined region.
  • There are no slopes in a constant function, as the rate of change (or gradient) is zero across the interval.
  • Constant functions do not depend on \(x\), other than defining where the line exists on the graph.
For plot nuances, it's crucial to mark the starting point with a filled circle at \((-1, 3)\) showing that \(x = -1\) is included.
Linear Function
Linear functions are identified by their straight-line graphs, and they have an equation of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. In the interval \(x > -1\), the function we are concerned with is \(f(x) = -x + 2\).For plotting, focus on:
  • Y-intercept: This is at \(y = 2\) when \(x = 0\).
  • Slope: The slope \(m = -1\) means for every increase in \(x\), \(y\) decreases by \(1\). This shows how steeply the line declines.
When graphing on the interval \(x > -1\), begin plotting from just right of \(x = -1\). Use an open circle at \((-1, 3)\) to denote exclusion from this function's values at that point. Maintain this slope pattern to extend the line through the positive side, characterizing its linearity.
Slope-Intercept Form
The slope-intercept form of a linear function is commonly written as \(y = mx + c\), where \(m\) represents the slope of the line, and \(c\) is the y-intercept, the location where the line crosses the y-axis. Understanding how this form shapes the graph is critical. Here's what you need to remember:
  • The slope \(m\) determines the line's inclination. A positive slope moves the line upwards as you move right, while a negative slope, like \(-1\) in \(-x + 2\), moves it downward.
  • The y-intercept \(c\) pinpoints the starting point on the y-axis. For instance, \(c = 2\) suggests the line crosses the y-axis at \(y = 2\).
When graphing in slope-intercept form, start at the y-intercept, and utilize the slope to determine further points. For every unit change in \(x\), adjust \(y\) accordingly by the slope. This framework makes constructing and interpreting these lines straightforward.

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Most popular questions from this chapter

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Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} x+2, & x<2 \\ 4, & x \geq 2 \end{array}\right.$$
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