Chapter 2.4, Problem 39AYU

(a)

To determine

To find: the domain of the given function.

Answer to Problem 39AYU

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

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=2\mathrm{int}\left(x\right)$

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 $f\left(x\right)$ is given by $2\mathrm{int}\left(x\right)$ . The greatest integer less than or equal to $x$ is multiplied by 2. This operation can be performed on any real number.

So, the domain of $x$ is the set of all the 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 39AYU

The $x$ intercept is $\left(x,0\right)$ for, $0\le x<1$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=2\mathrm{int}\left(x\right)$

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.

The value of the function $f\left(x\right)$ of the coordinate, when $0\le x<1$ is 0.

Therefore, the $x$ intercept is $\left(x,0\right)$ for, $0\le x<1$ .

(c)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=2\mathrm{int}\left(x\right)$

Calculation:

Graph the greatest integer function, $f\left(x\right)=2\mathrm{int}\left(x\right)$ . Substitute some values for $x$ in the equation to obtain few points on the graph.

$x$  $y=2\mathrm{int}\left(x\right)$$\left(x,y\right)$

1 2   $\left(1,2\right)$ $-2$ 4   $\left(-2,4\right)$ -1 $-2$$\left(-1,-2\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 39AYU

The range of $f$ is the set of even integers.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=2\mathrm{int}\left(x\right)$

Calculation:

The points on the graph of $f$ have all the $y$ coordinates as even integers. For each number $y$ , there is at least one number $x$ in the domain.

So, the range of $f$ is the set of even integers.

(e)

To determine

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

Answer to Problem 39AYU

The function $f$ is discontinuous on its domain.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=2\mathrm{int}\left(x\right)$

Calculation:

From the graph, it can be observed that there is a discontinuity at $x=0,1,2,\mathrm{...}$ . So, the function $f$ is discontinuous on its domain.