Chapter 2, Problem 69RE

(a)

To determine

To find: the domain of the given function.

Answer to Problem 69RE

The domain of the given function is $\left\{x|x\ge -4\right\}$ or $\left[-4,\infty \right)$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x\text{\hspace{1em}if}-4\le x<1\\ 1\text{\hspace{1em}if}x=1\\ 3x\text{\hspace{1em}if}x>0\end{array}$

Calculation:

The domain of the function $f\left(x\right)$ , is the set of all the possible values of $x$ .

The value of the function $f\left(x\right)$ when $-4\le x\le 0$ is given by $x$ .

These operations can be performed on real numbers between $-5$ and 0.

The value of the function for any value of $x=0$ is given by $1$ .

The value of the function $f\left(x\right)$ when $x>0$ is given by $3x$ . In the expression $3x$ , the value of the variable $x$ is multiplied by 3. This operation can be performed on real numbers greater than zero.

Therefore, the domain of the given function is $\left\{x|x\ge -4\right\}$ or $\left[-4,\infty \right)$ .

(b)

To determine

To locate: any intercepts of the given function.

Answer to Problem 69RE

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

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x\text{\hspace{1em}if}-4\le x<1\\ 1\text{\hspace{1em}if}x=1\\ 3x\text{\hspace{1em}if}x>0\end{array}$

Calculation:

The $x$ intercepts are those points for which $y$ coordinate is zero and the $y$ intercepts are those points for which the $x$ coordinate is zero.

Determine the points on the graph for which the $y$ coordinate when $x=0$ is 1.

So, $\left(0,1\right)$ is a point on the graph.

Therefore, the intercept is $\left(0,1\right)$ .

(c)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x\text{\hspace{1em}if}-4\le x<1\\ 1\text{\hspace{1em}if}x=1\\ 3x\text{\hspace{1em}if}x>0\end{array}$

Calculation:

Graph each piece to graph the function.

For plotting the graph of the line, $y=x$ , in the part for which $-4\le x<0$ , substitute various values for $x$ in the equation to obtain few points on the graph.

$x$  $f\left(x\right)=x$$\left(x,f\left(x\right)\right)$

$-4$ $-4$$\left(-4,-4\right)$   $-3$$-3$$\left(-3,-3\right)$   $-2$$-2$$\left(-2,-2\right)$   $-1$ $-1$ $\left(-1,-1\right)$

In order to plot the graph of the line $y=3x$ in the part for which $x>0$ , substitute various values for $x$ in the equation to obtain few points on the graph.

$x$  $f\left(x\right)=3x$$\left(x,f\left(x\right)\right)$

1   3 $\left(1,3\right)$ 2   6 $\left(2,6\right)$ 3   9 $\left(3,9\right)$

Plot the points and draw the lines to get the graph of the function.

  

(d)

To determine

To find: the range based on the graph.

Answer to Problem 69RE

The range of $f$ is $\left\{y|-4\le y<0ory>0\right\}$ or the interval $\left[-4,0\right)\cup \left(0,\infty \right)$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x\text{\hspace{1em}if}-4\le x<1\\ 1\text{\hspace{1em}if}x=1\\ 3x\text{\hspace{1em}if}x>0\end{array}$

Calculation:

From the graph, notice that the points on the graph of $f$ have the $y$ coordinate between $-4$ and 0 or $y$ greater than 0. For each number $y$ , there is at least one number $x$ in the domain.

So, the range of $f$ is $\left\{y|-4\le y<0ory>0\right\}$ or the interval $\left[-4,0\right)\cup \left(0,\infty \right)$ .

(e)

To determine

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

Answer to Problem 69RE

The function $f$ is discontinuous on its domain.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}x\text{\hspace{1em}if}-4\le x<1\\ 1\text{\hspace{1em}if}x=1\\ 3x\text{\hspace{1em}if}x>0\end{array}$

Calculation:

From the graph, it can be observed that there is a discontinuity at $x=0$ . So, the function $f$ is discontinuous on its domain.