Chapter 2.4, Problem 40AYU

(a)

To determine

To find: the domain of the given function.

Answer to Problem 40AYU

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

Explanation of Solution

Given information:

Given function

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

Calculation:

From the above definition we see the domain of the function is the set of all real numbers.

(b)

To determine

To locate: any intercepts of the given function.

Answer to Problem 40AYU

The $x$ intercepts are $\left(x,0\right)$ for $0\le x<\frac{1}{2}$ .

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

Explanation of Solution

Given information:

Given function

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

Calculation:

The intercepts are the points on the graph which are obtained when it cuts the $x$ and $y$ axes.

The $x$ intercepts are $\left(x,0\right)$ for $0\le x<\frac{1}{2}$ .

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

(c)

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function

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

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 40AYU

The range of the function $f\left(x\right)$ is the set of real numbers.

Explanation of Solution

Given information:

Given function

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

Calculation:

From the graph we see that $f\left(x\right)$ achieves all the real values in its domain. So the range of the function $f\left(x\right)$ is the set of all integers.

(e)

To determine

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

Answer to Problem 40AYU

The function is discontinuous at $\left\{x|x=\mathrm{i}\text{ntegral}\text{multiple}\text{of}\frac{1}{2}\right\}$

Explanation of Solution

Given information:

Given function

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

Calculation:

The graph shows that the function is discontinuous at all multiples of $\frac{1}{2}$ .

Therefore, the function is discontinuous at $\left\{x|x=\mathrm{i}\text{ntegral}\text{multiple}\text{of}\frac{1}{2}\right\}$ .