Precalculus Practice Test: Functions and Graphs
- Functions and Graphs
- 10 questions
- 15 minutes
These 10 questions review the function skills that the rest of precalculus builds on: domain and range, composition, inverses, symmetry and average rate of change. When you find a domain, look for two things first: division by zero and square roots of negative numbers.
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 domain of f(x) = x − 3x − 5?
Correct answer: C
The square root needs x − 3 ≥ 0, so x ≥ 3, and the denominator cannot be zero, so x ≠ 5. The interval [3, ∞) forgets to remove 5, and (3, 5) wrongly removes 3, where the function equals 0.
-
Question 2
Let f(x) = 2x + 1 and g(x) = x2 − 3. What is (f ∘ g)(2)?
Correct answer: A
First g(2) = 4 − 3 = 1, then f(1) = 2(1) + 1 = 3. The value 22 is g(f(2)) = g(5), which composes in the wrong order.
-
Question 3
What is the inverse of f(x) = 3x − 25?
Correct answer: D
Swap x and y: x = (3y − 2)/5, so 5x = 3y − 2 and y = (5x + 2)/3. The option with 5x − 2 undoes the subtraction with another subtraction instead of adding 2.
-
Question 4
Which function is odd?
Correct answer: B
For x3 − x, replacing x with −x gives −x3 + x = −f(x), so it is odd. The function x3 + 1 is neither odd nor even because the constant 1 does not change sign.
-
Question 5
What is the average rate of change of f(x) = x2 − 4x on the interval [1, 5]?
Correct answer: C
Here f(5) = 5 and f(1) = −3, so the rate is (5 − (−3)) ÷ (5 − 1) = 8 ÷ 4 = 2. The answer 8 is only the change in output; it still has to be divided by the change in x.
-
Question 6
What is the range of f(x) = −(x − 2)2 + 7?
Correct answer: A
The parabola opens downward and its vertex is (2, 7), so 7 is the largest output and every smaller value is reached. The interval (−∞, 2] uses the x-coordinate of the vertex instead of the y-coordinate.
-
Question 7
A function is defined in two pieces:
f(x) = x2 if x < 1
f(x) = 3x − 1 if x ≥ 1What is f(1) + f(−2)?
Correct answer: D
Since 1 ≥ 1, use the second piece: f(1) = 2. Since −2 < 1, use the first piece: f(−2) = 4. The sum is 6; the answer 5 comes from using x2 at x = 1.
-
Question 8
If f(x) = xx − 1 and f(a) = 3, what is the value of a?
Correct answer: B
Solve a = 3(a − 1): a = 3a − 3, so 2a = 3 and a = 3/2. Check: (3/2) ÷ (1/2) = 3. The value 3 is the output, not the input.
-
Question 9
Let f(x) = x and g(x) = x − 4. What is the domain of f(g(x))?
Correct answer: A
The composition is x − 4, which needs x − 4 ≥ 0, so x ≥ 4. The interval [0, ∞) is the domain of f alone, and (4, ∞) wrongly excludes 4, where the output is 0.
-
Question 10
Which function is one-to-one on all real numbers?
Correct answer: C
The cubic x3 + 2 is always increasing, so no horizontal line meets its graph twice. Each of the other functions repeats outputs; for example x2 gives 4 at both 2 and −2.
Questions you leave blank are not counted as wrong.
More Math tests
MathPrecalculus Practice Test: Function TransformationsTransformations10 questions15 min MathPrecalculus Practice Test: Polynomial FunctionsPolynomial Functions10 questions15 minMathPrecalculus Practice Test: Rational Functions and AsymptotesRational Functions and Asymptotes10 questions15 minMathPrecalculus Practice Test: Exponential FunctionsExponential 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