Excluded values and domain restrictions
Excluded Values and Domain Restrictions
Excluded Values and Domain Restrictions is a topic in Rational Functions in the California Common Core State Standards. It is aligned to Standard F-IF.5, which requires students to relate the domain of a rational function to its graph and the values it excludes.
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.