Chapter 11.1, Problem 45PPE

To determine

Write a rational expression that has 4 and − 3 as excluded values.

Answer to Problem 45PPE

  $\frac{x^2}{x^3-x^2-12x}$

Explanation of Solution

Given: Write a rational expression that has 4 and − 3 as excluded values.

Concept Used:

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

Excluded values are also called points of discontinuity. These are the values that make the denominator equal to zero and are not part of the domain.

In any rational function, the denominator cannot be zero.

To find the excluded value of the function, make denominator equal to zero and simplify.

Calculation:

Let the expression be: $\frac{x^2}{x^3-x^2-12x}$

Now simplify the expression and find the excluded values.

  $\frac{x^2}{x^3-x^2-12x}=\frac{x·x}{x(x^2-x-12)}=\frac{x·x}{x(x-4)(x+3)}=\frac{x}{(x-4)(x+3)}$

To find the excluded value of the function, make denominator equal to zero and simplify.

Excluded values: $\begin{array}{l}(x-4)(x+3)\ne 0\\ x-4\ne 0\Rightarrow x\ne 4\text{or}x+3\ne 0\Rightarrow x\ne -3\end{array}$

Thus, the rational expression $\frac{x^2}{x^3-x^2-12x}$that has 4 and − 3 as excluded values.