Chapter 2, Problem 68RE

(a)

To determine

To find: the domain of the given function.

Answer to Problem 68RE

The domain is $\left\{x|x\ge -3\right\}$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x-1\text{\hspace{1em}if}-3<x<0\\ 3x-1\text{\hspace{1em}if}x\ge 0\end{array}$

Calculation:

From the above definition we see the domain of the function is the set of all real numbers such that $x\ge -3$ .

Hence the domain is $\left\{x|x\ge -3\right\}$ .

(b)

To determine

To locate: any intercepts of the given function.

Answer to Problem 68RE

  $x$ intercept is $\left(\frac{1}{3},0\right)$ .

The $y$ intercept is $\left(0,-1\right)$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x-1\text{\hspace{1em}if}-3<x<0\\ 3x-1\text{\hspace{1em}if}x\ge 0\end{array}$

Calculation:

The points $\left(\frac{1}{3},0\right)$ and $\left(0,-1\right)$ satisfies the function above.

Hence,$x$ intercept is $\left(\frac{1}{3},0\right)$ .

The $y$ intercept is $\left(0,-1\right)$ .

(c)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x-1\text{\hspace{1em}if}-3<x<0\\ 3x-1\text{\hspace{1em}if}x\ge 0\end{array}$

Calculation:

Plot the points and draw the line to get the graph of the function

  

(d)

To determine

To find: the range based on the graph.

Answer to Problem 68RE

The range of the function $f\left(x\right)$ is the set $\left\{y|y>-4\right\}$.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x-1\text{\hspace{1em}if}-3<x<0\\ 3x-1\text{\hspace{1em}if}x\ge 0\end{array}$

Calculation:

From the graph we see that $f\left(x\right)>-4$ in its domain. So the range of the function $f\left(x\right)$ is the set $\left\{y|y>-4\right\}$.

(e)

To determine

To find: whether $f$ continuous on its domain or not.

Answer to Problem 68RE

The function is continuous at throughout its domain.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x-1\text{\hspace{1em}if}-3<x<0\\ 3x-1\text{\hspace{1em}if}x\ge 0\end{array}$

Calculation:

Consider the graph of the function.

  

The graph shows that the function is continuous at throughout its domain.