Chapter 11.3, Problem 93E

Solution

a)

To graph: The given function $y=\frac{x^3-1}{x-1}$ .

The graph for the function $y=\frac{x^3-1}{x-1}$ is:

  

Given information:

The given function is $y=\frac{x^3-1}{x-1}$ .

Calculation:

The given function is not defined when $x=1$ .

Plot the graph for the function $y=\frac{x^3-1}{x-1}$ such that $x\ne 1$ .

  

b)

To verify: That the function has one removable discontinuity.

The removal discontinuity is at $x=1$ .

Given information:

The given function is $y=\frac{x^3-1}{x-1}$ .

Calculation:

Factorize the given function.

  $\begin{array}{c}y=\frac{x^3-1}{x-1}\\ =\frac{\left(x-1\right)\left(x^2+x+1\right)}{x-1}\\ =x^2+x+1\end{array}$

Thus, the equation $y=x^2+x+1$ represents a parabola which is continuous for all real numbers.

Therefore, the removal discontinuity is $x=1$ .

c)

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

The formula for continuous extension of the function is $y=x^2+x+1$ .

Given information:

The given function is $y=\frac{x^3-1}{x-1}$ .

Calculation:

From part (b), it is verified that the equation can be represented as a parabola $y=x^2+x+1$ for all real values of real numbers except for $x=1$ .

Therefore, formula for continuous extension of the function is $y=x^2+x+1$ .