Chapter 2.4, Problem 30AYU

(a)

To determine

To find: the domain of the given function.

Answer to Problem 30AYU

The function is not defined.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}3x\text{if}x\ne 0\\ 4\text{if}x=0\end{array}$

Calculation:

From the above definition we see the domain of the function is the set of all real numbers as there is no value of $x$ where the function is not defined.

(b)

To determine

To locate: any intercepts of the given function.

Answer to Problem 30AYU

The $x$ intercepts is of length 0.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}3x\text{if}x\ne 0\\ 4\text{if}x=0\end{array}$

Calculation:

The intercepts are the points on the graph which are obtained when it cuts the $x$ and $y$ axes. The point $\left(0,4\right)$ lies on the graph. Hence the function has a $y$ intercept at $\left(0,4\right)$ .

Also, the value of $y$ becomes 0 only when the value of $x$ is 0. This means that the $x$ intercepts is of length 0.

(c)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}3x\text{if}x\ne 0\\ 4\text{if}x=0\end{array}$

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 30AYU

The range of $f$ is $\left\{y|y\ne 0\right\}$ or the interval $\left(-\infty ,0\right)\cup \left(0,\infty \right)$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}3x\text{if}x\ne 0\\ 4\text{if}x=0\end{array}$

Calculation:

From the graph we see the point $\left(0,0\right)$ do not lie on the graph. So the range of the function $f\left(x\right)$ is the set of real numbers except $y=0$ .

So, the range of $f$ is $\left\{y|y\ne 0\right\}$ or the interval $\left(-\infty ,0\right)\cup \left(0,\infty \right)$ .

(e)

To determine

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

Answer to Problem 30AYU

The function is discontinuous at $x=0$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}3x\text{if}x\ne 0\\ 4\text{if}x=0\end{array}$

Calculation:

From the graph, we see the point $\left(0,0\right)$ do not lie on the graph. Hence the function is discontinuous at $x=0$ .