CMClearMathAcademy

Domain Restrictions and Holes

A free Precalculus lesson from the “Rational Functions” unit, with a worked example and practice problems including step-by-step solutions.

A canceled factor creates a hole; an uncanceled denominator factor creates a vertical asymptote. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Rational functions model rates, constraints, efficiency, and quantities that change sharply near restricted inputs.

Worked example

Problem. For r(x) = (x - 3)/((x - 3)(x + 7)), find the x-value of the hole.

  1. A hole appears where a factor cancels from top and bottom.
  2. The factor (x - 3) cancels, so set x - 3 = 0.
  3. x = 3, so the hole is at x = 3.

Answer: 3

Practice problems

1. For r(x) = (x - 3)/((x - 3)(x + 7)), find the x-value of the hole.

Show solution
  1. A hole appears where a factor cancels from top and bottom.
  2. The factor (x - 3) cancels, so set x - 3 = 0.
  3. x = 3, so the hole is at x = 3.

Answer: 3

2. For r(x) = (3(x - 4))/(x - 4), cancel the shared (x - 4) factor and find the y-value of the hole.

Show solution
  1. The factor (x - 4) appears in both the numerator and the denominator, so cancel it.
  2. What remains is the constant 3, defined for every x except 4.
  3. The hole sits on that constant value, so its y-value is 3.

Answer: 3

3. For r(x) = ((x - 5)(x - 12))/(x - 5), the simplified rule is x - 12. Find the y-value of the hole (substitute x = 5).

Show solution
  1. Cancel (x - 5); the simplified rule is x - 12.
  2. The hole is at x = 5, so substitute: 5 - 12.
  3. 5 - 12 = -7, so the hole's y-value is -7.

Answer: -7

4. In r(x) = (x - 2)/((x - 2)(x + 7)), what occurs at x = 2?

Choices: a hole · a vertical asymptote · a horizontal asymptote · a zero of the function

Show solution
  1. The factor (x - 2) appears in both numerator and denominator.
  2. A factor that cancels produces a hole, not an asymptote.
  3. So x = 2 is a hole.

Answer: a hole

5. In r(x) = (x - 3)/((x - 3)(x + 8)), which factor cancels?

Choices: (x - 3) · (x + 8) · (x + 3) · (x - 8)

Show solution
  1. A factor cancels when it appears in both the numerator and the denominator.
  2. (x - 3) is in the numerator and in the denominator.
  3. So (x - 3) cancels.

Answer: (x - 3)

Practice this interactively with instant feedback and an AI tutor.

Practice Domain Restrictions and Holes Take the free placement check

More Precalculus lessons