Precalculus Practice Test: Rational Functions and Asymptotes
- Rational Functions and Asymptotes
- 10 questions
- 15 minutes
These 10 questions cover vertical, horizontal and slant asymptotes, holes, intercepts and rational equations and inequalities. Always factor first: a factor that cancels gives a hole, a factor that stays in the denominator gives a vertical asymptote.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
Your result
- 0 correct
- 0 wrong
- 0 blank
-
Question 1
What is the vertical asymptote of f(x) = x + 2x − 5?
Correct answer: B
The denominator is zero at x = 5 and the numerator is not zero there, so the line x = 5 is a vertical asymptote. At x = −2 the function has an x-intercept, not an asymptote.
-
Question 2
What is the horizontal asymptote of f(x) = 3x2 − 16x2 + x?
Correct answer: C
The degrees of the numerator and denominator are equal, so the asymptote is the ratio of leading coefficients: 3/6 = 1/2. The line y = 0 applies only when the denominator has the higher degree.
-
Question 3
Which feature does the graph of f(x) = x2 − 9x − 3 have at x = 3?
Correct answer: A
The expression simplifies to x + 3 for x ≠ 3, so the graph is a line with one missing point at (3, 3 + 3) = (3, 6). The factor cancels, so there is no vertical asymptote.
-
Question 4
What is the slant (oblique) asymptote of f(x) = x2 + 3x − 1x − 1?
Correct answer: D
Long division gives x2 + 3x − 1 = (x − 1)(x + 4) + 3, so f(x) = x + 4 + 3/(x − 1), and the last part goes to 0. The line y = x + 2 comes from dividing by x + 1 instead of x − 1.
-
Question 5
What is the horizontal asymptote of f(x) = 5xx2 + 1?
Correct answer: B
The denominator has the higher degree, so the fraction approaches 0 for large |x|. The value 5 would be the asymptote only if both degrees were equal.
-
Question 6
What are the x-intercepts of f(x) = x2 − 4x2 − x − 6?
Correct answer: A
Factor: (x − 2)(x + 2) / ((x − 3)(x + 2)). The factor x + 2 cancels, so x = −2 is a hole, not an intercept, and only x = 2 makes the function zero.
-
Question 7
What is the solution of 2x − 1 = 3x + 2?
Correct answer: C
Cross-multiplying gives 2(x + 2) = 3(x − 1), so 2x + 4 = 3x − 3 and x = 7. Neither denominator is zero at 7, so the solution is valid.
-
Question 8
Which function has vertical asymptotes x = 2 and x = −2 and the horizontal asymptote y = 0?
Correct answer: D
The denominator x2 − 4 is zero at ±2, nothing cancels, and the bottom degree is higher, so y = 0. With x − 2 on top, the factor cancels and x = 2 becomes a hole.
-
Question 9
What is the domain of f(x) = x + 1x2 + 5x + 6?
Correct answer: A
The denominator factors as (x + 2)(x + 3), which is zero at −2 and −3. The numerator does not affect the domain, so −1 is allowed.
-
Question 10
What is the solution set of x − 4x + 1 ≤ 0?
Correct answer: C
The quotient is negative between the critical values −1 and 4 and equals 0 at x = 4. The value −1 must be excluded because it makes the denominator zero.
Questions you leave blank are not counted as wrong.
More Math tests
MathPrecalculus Practice Test: Exponential FunctionsExponential Functions10 questions15 min MathPrecalculus Practice Test: Functions and GraphsFunctions and Graphs10 questions15 minMathPrecalculus Practice Test: Function TransformationsTransformations10 questions15 minMathPrecalculus Practice Test: Polynomial FunctionsPolynomial Functions10 questions15 minMathPrecalculus Practice Test: LogarithmsLogarithms10 questions15 minMathPrecalculus Practice Test: Unit Circle and Trig RatiosUnit Circle and Trig Ratios10 questions15 minMathPrecalculus Practice Test: Graphs of Trig FunctionsGraphs of Trig Functions10 questions15 min