Chapter 11.3, Problem 92E

Solution

a)

To graph: The given function $y=\frac{x-5}{5-x}$ .

The graph for the function $y=\frac{x-5}{5-x}$ is:

  

Given information:

The given function is $y=\frac{x-5}{5-x}$ .

Calculation:

The given function is not defined when $x=5$ .

Plot the graph for the function $y=\frac{x-5}{5-x}$ such that $x\ne 5$ .

  

b)

To verify: That the function has one removable discontinuity.

The removal discontinuity is at $x=5$ .

Given information:

The given function is $y=\frac{x-5}{5-x}$ .

Calculation:

Factorize the given function.

  $\begin{array}{c}y=\frac{x-5}{5-x}\\ =\frac{x-5}{-\left(x-5\right)}\\ =-1\end{array}$

Thus, the function is continuous for all real numbers except at $x=5$ .

Therefore, the removal discontinuity is $x=5$ .

c)

To find: The formula for continuous extension of the function.

The formula for continuous extension of the function is $y=-1$ .

Given information:

The given function is $y=\frac{x-5}{5-x}$ .

Calculation:

From part (b), it is verified that the equation can be represented as a horizontal line $y=-1$ for all real values of real numbers except for $x=5$

Therefore, formula for continuous extension of the function is $y=-1$ .