Problem 49 Graph: \(f(x)=\left\\{\begin{arr... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 2: Problem 49

Graph: $f(x)=\left\\{\begin{array}{ll}x-2, & \text { for } x \leq-1 \\ 3, & \text { for }-12\end{array}\right.$

Short Answer

Expert verified

The graph consists of three parts: a line $y = x - 2$ for $x \leq -1$, a horizontal line $y = 3$ for $-1 < x \leq 2$, and a line $y = x$ for $x > 2$.

Step by step solution

- Identify the Piecewise Function Intervals

The piecewise function is defined in three intervals. For each interval, identify the corresponding function:1. For $x \leq -1$, the function is $f(x) = x - 2$.2. For $-1 < x \leq 2$, the function is $f(x) = 3$.3. For $x > 2$, the function is $f(x) = x$.

- Graph the Interval $x \leq -1$

In this interval, $f(x) = x - 2$. This is a linear function with slope 1 and y-intercept -2. Plot a line starting from $(-(1, -3))$ to include points where $x \leq -1$. Make sure to use a solid dot at $x = -1$ since the function is equal to the expression at that point.

- Graph the Interval $-1 < x \leq 2$

In this interval, $f(x) = 3$. This is a constant function where $y$ is always 3. Draw a horizontal line at $y = 3$ from $x = -1$ to $x = 2$. Use an open circle at $x = -1$ and a solid dot at $x = 2$.

- Graph the Interval $x > 2$

In this interval, $f(x) = x$. This is a linear function with slope 1 passing through the origin. Start plotting a line from the point $(2, 2))$ with an open circle and draw it for all $x > 2$.

- Combine the Graphs

Combine the three pieces on the same coordinate plane. Ensure that each piece is drawn correctly according to its interval and the endpoints are marked properly with open or closed circles as necessary.

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Key Concepts

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

Piecewise functions

Piecewise functions are a type of function that is defined by multiple sub-functions, each corresponding to a part of the function's domain.
These sub-functions are sometimes called 'pieces'.
Each piece is defined over a specific interval and has its own rule.
For the given exercise, the piecewise function is defined as follows:

Graphing techniques

Graphing piecewise functions involves drawing each piece separately over its interval, then combining them on the same coordinate plane.
Make sure to handle each piece carefully, checking any open or closed circles which indicate the function's value at the interval endpoints:

For the interval $x \leq -1$, plot the line $f(x) = x - 2$ starting from $(-1, -3)$ with a solid dot at $x = -1$.
For the interval $-1 < x \leq 2$, draw a horizontal line $f(x) = 3$ from $x = -1$ to $x = 2$
For the interval $x > 2$, plot the line $f(x) = x$ starting at $(2,2)$ with an open circle and continuing right.

Interval notation

Interval notation is a mathematical notation used to describe subsets of the real number line.
In the context of piecewise functions, it helps define where each piece of the function is valid.
The notation uses brackets to indicate whether endpoints are included:

$[a, b]$ 鈥� includes both $a$ and $b$
$(a, b)$ 鈥� excludes $a$ and $b$
$(a, b]$ 鈥� excludes $a$, includes $b$
$[a, b)$ 鈥� includes $a$, excludes $b$

In our exercise:

For $x \leq -1$, use $( -\infty, -1 ]$
For $-1 < x \leq 2$, use $(-1, 2 ]$
For $x > 2$, use $(2, \infty)$

Function intervals

Each piece of the piecewise function corresponds to a specific interval on the x-axis:
The intervals divide the x-axis into segments where different rules apply.
This ensures a comprehensive graph that correctly represents the entire piecewise function.
Using the example provided:

For $x \leq -1$: $f(x) = x - 2$
For $-1 < x \leq 2$: $f(x) = 3$
For $x > 2$: $f(x) = x$

When graphing, make sure to mark the endpoints using open or closed circles accordingly:

Closed circles for $\leq$ or $\geq$
Open circles for $<$ or $>$

Start by plotting each piece, respecting these intervals, and then connect them to form the complete graph.

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91影视

Graph: \(f(x)=\left\\{\begin{array}{ll}x-2, & \text { for } x \leq-1 \\ 3, & \text { for }-12\end{array}\right.\)

Short Answer

Step by step solution

- Identify the Piecewise Function Intervals

- Graph the Interval \(x \leq -1\)

- Graph the Interval \(-1 < x \leq 2\)

- Graph the Interval \(x > 2\)

- Combine the Graphs

Key Concepts

Piecewise functions

Graphing techniques

Interval notation

Function intervals

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