ACT Math Practice Test: Intermediate Algebra and Functions
- Intermediate Algebra and Functions
- 12 questions
- 14 minutes
These 12 questions cover intermediate algebra and functions on the ACT: quadratics, systems of equations, function notation and composition, exponents and logarithms, rational expressions, sequences, absolute value and complex numbers. When an equation contains logarithms or variables in a denominator, check each solution in the original equation before you choose an answer.
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
The function f is defined by f(x) = 2x2 − 3x + 1. What is the value of f(−2)?
Correct answer: D
f(−2) = 2(−2)2 − 3(−2) + 1 = 2(4) + 6 + 1 = 15. Squaring −2 as −4 gives −1, and treating −3(−2) as −6 gives 3.
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Question 2
The functions f and g are defined by f(x) = 3x − 1 and g(x) = x2 + 2. What is the value of f(g(2))?
Correct answer: C
Work from the inside out: g(2) = 4 + 2 = 6, and then f(6) = 3(6) − 1 = 17. The choice 27 is g(f(2)), the composition in the opposite order, and 6 stops after the inner function.
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Question 3
2x + 3y = 7
4x − y = 21If (x, y) is the solution of the system of equations shown, what is the value of xy?
Correct answer: A
The second equation gives y = 4x − 21. Substituting, 2x + 3(4x − 21) = 7, so 14x = 70, x = 5 and y = −1. Then xy = −5; the choices 4 and 6 are x + y and x − y.
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Question 4
What are the solutions of the equation x2 − 6x + 2 = 0?
Correct answer: B
The quadratic formula gives x = (6 ± 36 − 8) ÷ 2 = (6 ± 27) ÷ 2 = 3 ± 7. The answer 3 ± 27 divides only the 6 by 2, and −3 ± 7 uses the wrong sign for −b.
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Question 5
What value of x satisfies the equation 2x + 3 = 8x − 1?
Correct answer: C
Write 8 as 23, so 8x − 1 = 23x − 3. Setting the exponents equal gives x + 3 = 3x − 3, so 2x = 6 and x = 3. Multiplying only the x by 3 (x + 3 = 3x − 1) leads to 2, and 6 is the value of 2x.
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Question 6
What are all values of x that satisfy the equation log3 x + log3 (x − 8) = 2?
Correct answer: B
The product rule gives log3 [x(x − 8)] = 2, so x(x − 8) = 32 = 9, x2 − 8x − 9 = 0 and x = 9 or x = −1. The logarithm of a negative number is undefined, so −1 is rejected. Adding the arguments instead of multiplying them (2x − 8 = 9) gives 172.
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Question 7
For x ≠ −3 and x ≠ 2, which of the following is equivalent to x2 − 9x2 + x − 6?
Correct answer: D
Factor both parts: x2 − 9 = (x − 3)(x + 3) and x2 + x − 6 = (x + 3)(x − 2). Cancelling the common factor x + 3 leaves x − 3x − 2. The expression −9x − 6 cancels the x2 terms, which is not allowed because they are terms, not factors.
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Question 8
In a geometric sequence, the 2nd term is 12 and the 5th term is 324. What is the 6th term?
Correct answer: C
From the 2nd term to the 5th term the common ratio is applied 3 times, so r3 = 324 ÷ 12 = 27 and r = 3. The 6th term is 324 × 3 = 972. Treating the sequence as arithmetic (adding 104 each time) gives 428, and using r = 27 ÷ 3 = 9 gives 2,916.
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Question 9
Which of the following describes all solutions of the inequality |2x − 5| ≤ 7?
Correct answer: A
The inequality means −7 ≤ 2x − 5 ≤ 7. Adding 5 gives −2 ≤ 2x ≤ 12, so −1 ≤ x ≤ 6. The answer x ≤ 6 ignores the lower bound, and "x ≤ −1 or x ≥ 6" solves |2x − 5| ≥ 7 instead.
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Question 10
For i = −1, which of the following is equal to (3 − 2i)(4 + i)?
Correct answer: D
Multiplying out gives 12 + 3i − 8i − 2i2 = 12 − 5i − 2(−1) = 14 − 5i. Treating i2 as 1 gives 10 − 5i, and 12 − 2i multiplies only the real parts together and the imaginary coefficients together.
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Question 11
A ball is thrown upward from a platform. Its height, in feet, t seconds after it is thrown is given by h(t) = −16t2 + 64t + 5. What is the maximum height of the ball, in feet?
Correct answer: B
The maximum occurs at the vertex, t = −64 ÷ (2 × (−16)) = 2 seconds, and h(2) = −64 + 128 + 5 = 69 feet. The choice 133 leaves out the −16t2 term, and 197 adds that term instead of subtracting it.
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Question 12
The function f is defined by f(x) = 2x − 53. Which of the following defines the inverse function f−1?
Correct answer: A
Write y = 2x − 53, swap x and y, and solve: 3x = 2y − 5, so y = 3x + 52. The expression 32x − 5 is the reciprocal of f(x), which is not the same as the inverse function.
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