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Rational Functions and Asymptotes

A free College Algebra lesson from the “Advanced Function Types” unit, with a worked example and practice problems including step-by-step solutions.

A rational function is a ratio of polynomials. Its graph can have vertical asymptotes where the denominator is zero and horizontal or slant asymptotes based on end behavior. These features matter because they describe where the function is undefined and how the graph behaves far from the origin. When practicing, factor the numerator and denominator first, identify values that make the denominator zero, and check whether any factors cancel to create holes instead of asymptotes. A common mistake is calling every excluded value an asymptote without checking for cancellation.

What you'll learn

Why it matters: Reciprocal models, rate limits, concentration formulas, and average-cost curves can approach values they never reach or break at restricted inputs. Asymptotes make that behavior visible instead of treating it as a calculator error.

Worked example

Problem. For f(x) = 1/(x - 6), what is the vertical asymptote?

  1. Set the denominator equal to zero.
  2. x - 6 = 0.
  3. The vertical asymptote is x = 6.

Answer: x = 6

Practice problems

1. For f(x) = 1/(x + 3), what x-value is the vertical asymptote?

Show solution
  1. Warm-up: First identify exactly what the question is asking: For f(x) = 1/(x + 3), what x-value is the vertical asymptote?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. x + 3 = 0.
  4. Check the result by substituting or estimating: the response should match -3 and make sense in the original problem.

Answer: -3

2. For f(x) = 5/(x - 9), what x-value is the vertical asymptote?

Show solution
  1. Core Practice: First identify exactly what the question is asking: For f(x) = 5/(x - 9), what x-value is the vertical asymptote?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. x - 9 = 0.
  4. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

3. A vertical asymptote describes behavior near a restricted...

Choices: x-value · y-value only · slope · degree

Show solution
  1. Challenge: First identify exactly what the question is asking: A vertical asymptote describes behavior near a restricted...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. It is a vertical line x = a.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x-value

4. What is the vertical asymptote of f(x) = 1/(x - 4)? Enter the x-value.

Show solution
  1. Vertical Asymptotes: First identify exactly what the question is asking: What is the vertical asymptote of f(x) = 1/(x - 4)? Enter the x-value.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The denominator is zero at x = 4.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

5. What is the horizontal asymptote of f(x) = 3/(x + 1)? Enter the y-value.

Show solution
  1. Horizontal Asymptotes: First identify exactly what the question is asking: What is the horizontal asymptote of f(x) = 3/(x + 1)? Enter the y-value.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The denominator degree is larger than the numerator degree.
  4. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

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