GRE Quantitative Reasoning Practice: Functions, Quadratics and Inequalities
- Functions, Quadratics and Inequalities
- 10 questions
- 15 minutes
This set goes beyond basic equations to the algebra the GRE calls Advanced Math: functions, quadratic equations, absolute value and lines. Substituting the answer choices back into the question is a fast way to check your work.
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.
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Question 1
f(x) = 2x² − 3x + 1
What is the value of f(−2)?
Correct answer: E
f(−2) = 2(4) − 3(−2) + 1 = 8 + 6 + 1 = 15. A sign slip on the middle term, using −6 instead of +6, gives 3.
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Question 2
Which of the following describes all values of x for which |2x − 5| < 7?
Correct answer: A
|2x − 5| < 7 means −7 < 2x − 5 < 7. Adding 5 gives −2 < 2x < 12, and dividing by 2 gives −1 < x < 6. Answer C drops the lower bound, which absolute value always creates.
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Question 3
The roots of x² − 5x − 14 = 0 are r and s.
What is the value of r² + s²?
Correct answer: D
x² − 5x − 14 = (x − 7)(x + 2), so the roots are 7 and −2, and 49 + 4 = 53. Squaring the sum of the roots (5² = 25) is not the same as adding the squares.
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Question 4
f(x) = 3x − 4 and g(x) = x² + 1
What is the value of f(g(2))?
Correct answer: B
g(2) = 5, and f(5) = 15 − 4 = 11. Applying the functions in the other order, g(f(2)) = g(2) = 5, gives a different value.
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Question 5
f(x) = 2x + 35
If f(a) = 7, what is the value of a?
Correct answer: C
Set (2a + 3)/5 = 7, so 2a + 3 = 35, 2a = 32 and a = 16. The value 3.4 is f(7), which answers the reverse question.
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Question 6
What is the minimum value of y = x² − 6x + 13?
Correct answer: C
Completing the square, y = (x − 3)² + 4. The square is never negative, so the minimum is 4, reached at x = 3. The value 3 is where the minimum occurs, not the minimum itself.
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Question 7
3x + 2y = 16 and x − y = 2
What is the value of xy?
Correct answer: D
From the second equation, x = y + 2. Substituting: 3(y + 2) + 2y = 16, so 5y = 10, y = 2 and x = 4. Then xy = 8.
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Question 8
For how many integers x is x² < 20?
Correct answer: B
x² < 20 holds for −4, −3, −2, −1, 0, 1, 2, 3 and 4, which is 9 integers, since 5² = 25 is too large. Leaving out 0 gives 8, and leaving out the negative integers gives 5.
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Question 9
x² − y² = 36 and x + y = 9
What is the value of x?
Correct answer: E
Factor: (x + y)(x − y) = 36, so x − y = 36 ÷ 9 = 4. Adding x + y = 9 and x − y = 4 gives 2x = 13, so x = 6.5. The value 4 is x − y, and 2.5 is y.
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Question 10
A line passes through the points (2, 5) and (6, 13).
At what value of x does the line cross the x-axis?
Correct answer: A
The slope is (13 − 5)/(6 − 2) = 2, so the line is y = 2x + 1. Setting y = 0 gives x = −12. The value 1 is the y-intercept and 2 is the slope.
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