Explain why a number(s) must be excluded from the domain of a rational expression and how to determine which numbers must be excluded. Please provide an example with your explanation.

We want to see why in rational expressions we need to remove some elements from the domain.

This is done because we want to avoid undefined terms.

The first thing that you need to remember, is that the division by zero is undefined.

Then when we have a rational expression like:

All the values such that:

g(x) = 0 and f(x) ≠ 0.

need to be removed from the domain, because these values would cause us to have a zero in the denominator.

An example of this is:

Where we need to remove x = 7 from the domain in order to avoid a zero in the denominator.

If you want to learn more, you can read:

link

Step-by-step explanation:

"Rational expression" is a term generally applied to the ratio of two functions of an independent variable. If the denominator function is zero, the rational expression has the value "undefined." The domain of a function is the set of values for which the function is defined, so the domain (by definition) excludes variable values that make the function "undefined." That is, denominator zeros are excluded.

Consider the simple rational expression ...

  f(x) = 1/(x-1)

The denominator will be zero when x=1, so that value of x is excluded from the domain of f(x). The domain of f(x) is all real numbers except x=1.

To determine what numbers are excluded from the domain, find the zeros of the denominator of the rational function. If the rational expression is a compound fraction, find the zeros of any denominator, whether or not they are cancelled by simplification.

Example: (1/(x-2))/(2/((x-1)(x-2)) = (x-1)(x-2)/(2(x-2)) = (x-1)/2

The domain must exclude both x=1 and x=2 because those are zeros of the denominators in the original expression.

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