Question

How do you state the excluded values for rational expressions?

Answer 1

Hint: Given the rational expression. We have to determine the excluded values for rational expressions. First, we have to determine the values that will make the denominator equal to zero by equating the denominator expression equal to zero and find the value of the variable.

Complete step by step solution:
Given the rational expression in the form, $\dfrac{{f\left( x \right)}}{{g\left( x \right)}}$

The excluded values of the expression are the values for which $g\left( x \right)$ must be equal to zero. These values are also known as points of discontinuity.

The excluded values can be determined by equating the expression at the denominator equal to zero. Then, find the factors of the expression.

Then, set each factor equal to zero and find the value of the variable.

The value of the variable thus obtained are the excluded values for which denominator will be zero, and make the rational expression undefined.

For example, consider the rational expression, $\dfrac{{{x^2} - 5x + 6}}{{{x^2} - 2x}}$

Find the excluded value of the expression, equate the denominator expression equal to zero.

$ \Rightarrow {x^2} - 2x = 0$

Now, take out the common term from the left hand side of the expression.

$ \Rightarrow x\left( {x - 2} \right) = 0$

Now, we will set each factor equal to zero.

$ \Rightarrow x = 0{\text{ or }}\left( {x - 2} \right) = 0$

Simplify the expression.

$ \Rightarrow x = 0{\text{ or }}x = 2$

Therefore, the values $x = 0$ and $x = 2$ are called excluded values, which cannot be included in the domain of the rational expression.

Note: Please note that the value at the denominator must not be equal to zero, which makes the rational expression undefined or in the divide by zero forms. In such types of questions, students must identify such values and exclude them from the domain while solving the questions.