Rational Expressions and When They Are Undefined
A rational expression is undefined wherever its denominator equals zero, because division by zero has no answer. Factor the bottom to find every x-value that breaks the fraction.
A fraction is just division, and division by zero is not a number, not even infinity. So the instant the denominator of a rational expression hits zero, the expression has no value at that input. That's not a graphing glitch, it's a hard rule of arithmetic. The job is to find every x that makes the bottom zero before you do anything else to the expression.
To find those x-values, factor the denominator completely and set each factor equal to zero. Each solution is a value x can never take. This matters even after you simplify the expression, because simplifying happens after the restriction is already locked in — the original denominator decides what's excluded, not whatever is left once you've cancelled.
This connects to graphing rational functions: every excluded x shows up either as a vertical asymptote or as a hole, depending on whether that factor also cancels from the numerator. It also sets up the next step in this unit, solving rational equations, where you'll clear denominators to solve but still have to check your answers against these same excluded values before accepting them.
Key terms
- Excluded value
- An x-value that makes the original denominator zero, so it can never be plugged into the expression.
- Removable hole
- A single missing point on a graph where a factor cancelled from both top and bottom.
- Vertical asymptote
- A vertical line the graph approaches but never touches, from a factor left only in the denominator.
- Factoring
- Rewriting an expression as a product of simpler terms, used here to expose every zero of the denominator.
Watch it explained
- How it worksHow does factoring the bottom of a fraction reveal every value that has to be thrown out?
- Worked exampleFind every value of x for which (x + 2)/(x2 - 9) is undefined.
- Worked exampleWalk me through simplifying (x2 - 4)/(x2 + 5x + 6) and stating the excluded values.
- Common mistakeWhy isn't a cancelled factor allowed to erase the hole it leaves behind in the graph?
- Going furtherWhat happens when the top and the bottom of a fraction both hit zero at the same x?
Check yourself
End of topic.