Excluded values and domain restrictions
Excluded Values and Domain Restrictions
Excluded Values and Domain Restrictions is a topic in Rational Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.6(J), which requires students to determine the values of the domain for which a rational function is undefined.
Excluded values make a denominator zero; a cancelled factor leaves a hole, an uncancelled one a vertical asymptote.
Theory
An excluded value makes a denominator zero, so the function is undefined there:
- Set each denominator factor equal to zero.
- A factor that cancels with the numerator leaves a hole.
- A factor that does not cancel gives a vertical asymptote.
Excluded values:
How to find excluded values
- Factor the denominator.
- Set each factor equal to zero.
- Those \(x\)-values are excluded.
- Decide hole (cancels) vs asymptote (does not).
Set the denominator to zero.
| \(x-3\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(3\ \text{(excluded)}\) |
Factor and set each factor to zero.
| \(x^2-4\) | \(=\) | \((x-2)(x+2)\) |
| \(x\) | \(=\) | \(2,\ -2\) |
Factor the denominator.
| \(x^2+x-6\) | \(=\) | \((x+3)(x-2)\) |
| \(x\) | \(=\) | \(-3,\ 2\) |
Exclude \(x=3\).
| \(\text{domain}\) | \(=\) | \((-\infty,3)\cup(3,\infty)\) |
Common pitfalls
Frequently asked questions
What is an excluded value?
An \(x\)-value that makes a denominator zero, where the function is undefined.
How do you find excluded values?
Set each denominator factor equal to zero.
What is the difference between a hole and an asymptote?
A cancelled factor gives a hole; an uncancelled one gives a vertical asymptote.
Do excluded values stay after simplifying?
Yes — the domain restriction remains even after cancelling.