What is a domain?

**With rational expressions, the domain is the set of all values for the value in which evaluating the expression will not create dividing by zero.**

In what circumstances would substituting a variable into a rational expression result in division by zero?

Here are some examples.

**Ex. 1**

\(\dfrac{4}{x}\)

We know that we cannot divide by zero. What values of \(x\) would create division by zero? In this situation, the only value that would cause division of zero is \(x=0\). Therefore the domain of this rational expression is all real numbers except for 0.

**Ex. 2**

\(\dfrac{x}{4}\)

Again, we know we can’t divide by zero. So, does it matter that the variable is only in the numerator? We can divide into zero, just not by 0. Since, there is no variable, the domain is all real numbers.

**Ex. 3**

\(\dfrac{x+4}{x^2+7x+10}\)

It is hard to tell which values of \(x\) will make the denominator zero without factoring the trinomial. The trinomial in the denominator factors to \((x+2)(x+5)\). Therefore, \(x\) cannot be \(-2\) or \(-5\). Either would make the denominator zero as \((x+2)\) and \((x+5)\) are factors. Any number times zero is equal to zero. The domain is all real numbers except \(-2\) and \(-5\).

Why is \(-4\) an acceptable value for \(x\)?

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