A rational function is undefined wherever its denominator is zero. The graph breaks there.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
That break has a name.
Vertical asymptote is a line the graph approaches but never reaches, where the function is undefined.
Solving
Multiply through by the common denominator to clear the fractions, then solve as usual.
Why you must check
Multiplying by an expression that could be zero can create answers that do not work. Always test them.
The domain excludes zeros of the denominator
A rational function is undefined wherever its denominator is zero. Finding those values is the first step in any rational problem, and they remain excluded even if a factor later cancels.
Asymptote or hole
A zero of the denominator that does not cancel gives a vertical asymptote. One that cancels with the numerator gives a hole — a single missing point. The distinction is decided by factoring, not by inspection.
End behaviour from the degrees
If the numerator's degree is lower, the horizontal asymptote is y = 0. If equal, it is the ratio of leading coefficients. If higher, there is no horizontal asymptote — the function grows without bound.
Extraneous solutions
Multiplying through by a denominator can introduce solutions that make that denominator zero. Every answer must be checked against the original domain, and one that was excluded from the start must be discarded.
Step 2: Try It Yourself
The curve breaks at the asymptote rather than crossing it.
Step 3: Watch an Example
One step at a time.
Watch Ravi Reject an Answer
Ravi solves 1/(x − 2) = 3/(x − 2) − 1.
- Step 1
He multiplies through by (x − 2) to clear the fractions.
Step 4: Your Turn
Practice makes it stick.
The Asymptote
Problem 1 of 2
y = 1/(x − 5). At what x is it undefined?
The Solve
Problem 2 of 2
Solve 6/x = 3.
Breaks and Checks
1 of 4
y = 1/(x + 3). At what x is it undefined?
2 of 4
Solve 10/x = 5.
3 of 4
Solve 12/x = 4.
4 of 4
Must you check answers to a rational equation?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = 1/(x − 7). At what x is it undefined?
Question 2 of 2
Why can a rational equation produce an answer that does not work?
What You Learned
- A rational function breaks where its denominator is zero.
- Clear the denominators to solve.
- Always check: some answers are extraneous.