Chapter 11.2, Problem 58E

Solution

a)

To draw: The graph of $f$ . Find the domain and range.

The graph of the function $f$ is shown in Figure (1).

Domain of $f$ is $ℝ-\left\{2\right\}$

Range of $f$ is $\left\{1\right\}\cup \left(2,\infty \right)$ .

Given information:

The function $f\left(x\right)=\left\{\begin{array}{l}1\text{ if }x<2\\ x\text{ if }x>2.\end{array}\right.$

  $$

  $$

Calculation:

Graph of function:

The function is constant (equal to 1) for all the values less than 2 and for the values greater than 2, the function is $y=x$ . The graph is shown in Figure (1).

  

Figure (1)

Domain of the function:

The function $f$ is defined for all real numbers except $x=2$ . Thus, the domain of $f$ is $ℝ-\left\{2\right\}$ .

Range of the function:

By observing the Figure (1), the function can assume the values 1 and all real numbers greater than 2. Thus, the range of the function $f$ is $\left\{1\right\}\cup \left(2,\infty \right)$ .

b)

To find:define the area of the function from $x=0$ to $x=2$ .

No. There is no difference in the area of $f$ if the function has no value at $x=2$ .

Given information:

The function $f\left(x\right)=\left\{\begin{array}{l}1\text{ if }x<2\\ x\text{ if }x>2.\end{array}\right.$

  $$

Formula used:

  ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ = $\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}f\left(a+i\Delta x\right)\Delta x}$ , $\Delta x=\frac{b-a}{n}$

Calculation:

  ${\displaystyle {\int }_0^{2}f\left(x\right)dx=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}1\left(\frac{2}{n}\right)}}$

  $=2$

Thus, approximating the Reimann sum using left-end points, there is no difference in the area of under $f$ from $x=0$ to $x=2$ .