Chapter 2.3, Problem 31E

To determine

To find: The reason of the continuity of the function $f\left(x\right)=\frac{1}{x-3}$ .

Answer to Problem 31E

The function $f\left(x\right)=\frac{1}{x-3}$ is continuous for all the points of the domain of the function. So, the function $f\left(x\right)=\frac{1}{x-3}$ is continuous function.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{1}{x-3}$ .

Calculation:

As the function $f\left(x\right)=\frac{1}{x-3}$ is not defined for $x=3$ as the denominator becomes zero. So, the domain of the function $f\left(x\right)=\frac{1}{x-3}$ is all the values of $x$ except $x=3$ .

Check the continuity of the function $f\left(x\right)=\frac{1}{x-3}$ with the help of graph for the domain of the function.

To graph a function $y=\frac{1}{x-3}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{1}{x-3}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{3}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=1/\left(\text{X}-3\right)$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown in Figure (1).

  

Figure(1)

As observed from the graph, the function $f\left(x\right)=\frac{1}{x-3}$ is continuous for all the values of $x$ except $x=3$ .

So, the function $f\left(x\right)=\frac{1}{x-3}$ is continuous for all the values of domain.

Therefore, the function $f\left(x\right)=\frac{1}{x-3}$ is continuous function.