Chapter 2.4, Problem 29AYU

(a)

To determine

To find: the domain of the given function.

Answer to Problem 29AYU

The domain of the given function is the set of all the real numbers.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}2x\text{if}x\ne 0\\ 1\text{if}x=0\end{array}$

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 $f\left(x\right)$ at $x=0$ is 1. So, $f\left(0\right)=1$ . The value of the function for any value of $x$ , other than 0 is given by $2x$ .

In the expression $2x$ , the value of the variable $x$ is multiplied b 2. These operations can be performed on any real number.

So, the domain of $x$ is the set of all real numbers.

Therefore, the domain of the given function is the set of all the real numbers.

(b)

To determine

To locate: any intercepts of the given function.

Answer to Problem 29AYU

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

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}2x\text{if}x\ne 0\\ 1\text{if}x=0\end{array}$

Calculation:

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

The value of the function $f\left(x\right)$ or the $y$ coordinate is 1 when $x=0$ .

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

(c)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}2x\text{if}x\ne 0\\ 1\text{if}x=0\end{array}$

Calculation:

Graph given piece to graph the function.

For plotting the graph of the line $y=2x$ , substitute various for $x$ in the equation to obtain some points on the graph.

$x$	$y=2x$	$\left(x,y\right)$
1	2	$\left(1,2\right)$
$-1$	$-2$	$\left(-1,-2\right)$

The points $\left(0,1\right)$ is also on the graph because, when $x=0,f\left(x\right)=1$ .

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 29AYU

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

  $f\left(x\right)=\{\begin{array}{l}2x\text{if}x\ne 0\\ 1\text{if}x=0\end{array}$

Calculation:

From the graph, notice that the points on the graph of $f$ have the $y$ coordinates between 3 and $-3$ , excluding 0. For each number $y$ , there is at least one number $x$ in the domain.

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 29AYU

The function $f$ is disconrinous on its domain.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\{\begin{array}{l}2x\text{if}x\ne 0\\ 1\text{if}x=0\end{array}$

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.