Chapter 4, Problem 21CR

For the function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

Find the domain of $f$.

Locate any intercepts.

Graph the function.

Based on the graph, find the range.

To determine

The domain of $f$, where $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

Answer to Problem 21CR

Solution:

The domain of $f$ is $\left(-3,\infty \right)$

Explanation of Solution

Given information:

The function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

Explanation:

Here, the function $2x+1$ is a polynomial function on the interval $\left(-3,2\right)$, and $-3x+4$ is also a polynomial function on interval $[2,\infty )$

Thus, the function is defined on the interval $\left(-3,2\right)\cup [2,\infty )=\left(-3,\infty \right)$

Therefore, the domain of the function is $\left(-3,\infty \right)$

To determine

To graph: The intercepts of the function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

Explanation of Solution

Given information:

The function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

Graph:

First, find the $x$ - intercepts of the function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

First, find the intercepts for the function on interval $-3<x<2$ for that substitute $f\left(x\right)=0$ in the function and solve for $x$

By substituting $f\left(x\right)=0$ in $f\left(x\right)=2x+1$, it gives

$0=2x+1$

$\Rightarrow 2x=-1$

$\Rightarrow x=\frac{-1}{2}=-0.5$

Hence, the $x$ intercept is $-0.5$, and the point on the graph is $\left(-0.5,0\right)$

Now, find the $y$ - intercepts of the function $f\left(x\right)=2x+1$ for that substitute $x=0$ in the function and solve for $x$

By substituting $x=0$ in $f\left(x\right)=2x+1$

$f\left(x\right)=2\left(0\right)+1$

$\Rightarrow f\left(x\right)=1$

Hence, the $y$-intercept is $1$, and the point on the graph is $\left(0,1\right)$

Now, find the intercepts for the function on interval $x\ge 2$ for that substitute $f\left(x\right)=0$ in the function and solve for $x$

By substituting $f\left(x\right)=0$ in $f\left(x\right)=-3x+4$

$0=-3x+4$

$\Rightarrow 3x=4$

$\Rightarrow x=\frac{4}{3}=1.33333$

Since, $x\ge 2$ so $x=1.33333$ is not the $x$-intercept in given interval.

Now, find the $y$ - intercepts of the function $f\left(x\right)=-3x+4$ for that substitute $x=0$ in the function and solve for $x$.

Since, here $x=0$, this is less than two, so it does not lie in the given interval.

Hence, the intercepts to the function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$ are

Now, locate points $\left(-0.5,0\right),\left(0,1\right)$

The graph is as follows:

Interpretation:

The graph represents the intercepts of the function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

To determine

To graph: The function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

Explanation of Solution

Given information:

The function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

Graph:

From part (b)

The intercepts of the function on interval $-3<x<2$ are $\left(-0.5,0\right),\left(0,1\right)$ draw a line passing through these points and restrict this line on interval $\left(-3,2\right)$

Now, to graph the function on the interval $x\ge 2$, take random values of $x$ and find the corresponding value of $f\left(x\right)$ as follows

For $x=2$, the value of $f\left(x=2\right)=-3\left(2\right)+4=-2$, thus the point $\left(2,-2\right)$

For $x=3$, the value of $f\left(x=3\right)=-3\left(3\right)+4=-5$, thus the point $\left(3,-5\right)$

For $x=4$, the value of $f\left(x=4\right)=-3\left(4\right)+4=-8$, thus the point $\left(4,-8\right)$

Now, plot the points $\left(2,-2\right),\left(3,-5\right),\left(4,-8\right)$ and draw a line passing through these points for $x\ge 2$.

The graph of the function is as follows:

Interpretation:

There is break in the graph of function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$ at $x=2$

To determine

The range of the function from the graph of function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$.

Answer to Problem 21CR

Solution:

The range of the function is $\left(-\infty ,5\right)$

Explanation of Solution

Given information:

The function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$

Explanation:

From part (c), the graph of the function $f\left(x\right)=\{\begin{array}{c}2x+1\text{if}-3<x<2\\ -3x+4\text{if}x\ge 2\end{array}$ is as follows:

By observing the graph, the value of the function is less than $5$, that is $f\left(x\right)<5$

Hence, the range of the function is $\left(-\infty ,5\right)$