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116 Precalculus Algebra Practice Exam
1. For f(x) = 3x + 2, what is the function’s domain? A. All real numbers B. x ≥ 0 C. x > 0 D. x < 0 Answer: A. All real numbers Explanation: A linear function is defined for every real number. 2. If f(x) = x² and g(x) = 2x + 1, what is (f + g)(x)? A. x² + 2x + 1 B. x² + 2x C. 2x² + 2x + 1 D. x² + 1 Answer: A. x² + 2x + 1 Explanation: (f + g)(x) = f(x) + g(x) = x² + (2x + 1). 3. Given h(x) = x², what is h(–x)? A. x² B. – x² C. – x D. x Answer: A. x² Explanation: Squaring – x yields (–x)² = x². 4. What is the inverse of f(x) = x + 5? A. f⁻¹(x) = x – 5 B. f⁻¹(x) = 5 – x C. f⁻¹(x) = 1/(x + 5) D. f⁻¹(x) = – x + 5 Answer: A. f⁻¹(x) = x – 5 Explanation: Solving y = x + 5 for x gives x = y – 5; thus, f⁻¹(x) = x – 5. 5. What is the range of f(x) = √x? A. [0, ∞) B. (–∞, ∞) C. (0, ∞) D. (–∞, 0] Answer: A. [0, ∞) Explanation: The square root function only produces nonnegative outputs. 6. If f(x) = 2x and g(x) = x + 3, what is (f ∘ g)(x)? A. 2x + 3 B. 2x + 6 C. 2x + 9
D. 2x + 5 Answer: C. 2(x + 3) = 2x + 6 (Note: Option C must read “2x + 6”) Explanation: (f ∘ g)(x) = f(g(x)) = 2(x + 3) = 2x + 6.
7. Is the function f(x) = x³ even, odd, or neither? A. Even B. Odd C. Neither D. Both even and odd Answer: B. Odd Explanation: Since f(–x) = – x³ = – f(x), the function is odd. 8. Given the piecewise function f(x) = { x + 2 for x < 0, x² for x ≥ 0 }, what is f(–1)? A. 1 B. – 1 C. 0 D. –? Answer: A. 1 Explanation: For x = – 1 (which is less than 0), f(–1) = (–1) + 2 = 1. 9. The graph of f(x) = (x – 3)² is a translation of which graph? A. y = x² shifted right 3 units B. y = x² shifted left 3 units C. y = x² shifted up 3 units D. y = x² reflected about the y-axis Answer: A. y = x² shifted right 3 units Explanation: Replacing x by (x – 3) shifts the graph 3 units to the right. 10. What is the domain of f(x) = 1/(x – 4)? A. All real numbers except x = 4 B. x > 4 C. x < 4 D. All real numbers Answer: A. All real numbers except x = 4 Explanation: The function is undefined when the denominator is zero, so x ≠ 4. 11. For f(x) = 2x, what is f(f(x))? A. 4x B. 2x + 2 C. 2x² D. 4x + 1 Answer: A. 4x Explanation: f(f(x)) = f(2x) = 2(2x) = 4x. 12. Determine the inverse of f(x) = (x – 1)/3. A. f⁻¹(x) = 3x + 1 B. f⁻¹(x) = 3x – 1
B. 4
C. – 4
D. 2
Answer: A. 0 Explanation: f(2) = 2² – 4 = 4 – 4 = 0.
19. If f(x) = 3x + 2, what is f(1)? A. 5 B. 3 C. 2 D. 6 Answer: A. 5 Explanation: f(1) = 3(1) + 2 = 5. 20. Find the inverse of f(x) = 5 – x. A. f⁻¹(x) = 5 – x B. f⁻¹(x) = x – 5 C. f⁻¹(x) = – x + 5 D. f⁻¹(x) = 5 + x Answer: A. f⁻¹(x) = 5 – x Explanation: The function is symmetric; its inverse is itself. 21. What is the domain of f(x) = 1/√x? A. x > 0 B. x ≥ 0 C. All real numbers D. x ≠ 0 Answer: A. x > 0 Explanation: The square root requires nonnegative x, but the denominator cannot be zero. 22. If f(x) = x + 1 and g(x) = 2x, what is (g ∘ f)(x)? A. 2x + 2 B. 2x + 1 C. 2x + 3 D. 2x Answer: A. 2(x + 1) = 2x + 2 Explanation: (g ∘ f)(x) = g(f(x)) = 2(x + 1). 23. Which of the following functions is even? (A) f(x) = x², (B) f(x) = x³, (C) f(x) = x + 1, (D) f(x) = √x A. f(x) = x² B. f(x) = x³ C. f(x) = x + 1 D. f(x) = √x Answer: A. f(x) = x² Explanation: Since (–x)² = x², the function is even.
24. What is the range of f(x) = |x|? A. [0, ∞) B. (–∞, ∞) C. (0, ∞) D. [–∞, 0] Answer: A. [0, ∞) Explanation: Absolute value outputs are always nonnegative. 25. For f(x) = 2(x + 3), what is f(–3)? A. 0 B. – 6 C. 6 D. – 3 Answer: A. 0 Explanation: f(–3) = 2(–3 + 3) = 2(0) = 0. 26. What is the degree of the polynomial f(x) = 4x³ – 2x + 1? A. 3 B. 2 C. 1 D. 0 Answer: A. 3 Explanation: The highest exponent on x is 3. 27. Find a zero of f(x) = x² – 9. A. 3 B. – 3 C. Both 3 and – 3 D. 0 Answer: C. Both 3 and – 3 Explanation: x² – 9 factors as (x – 3)(x + 3). 28. What is the end behavior of f(x) = – x⁴ + 2x²? A. f(x) → – ∞ as |x| → ∞ B. f(x) → ∞ as x → ∞ C. f(x) → ∞ as x → – ∞ D. f(x) remains bounded Answer: A. f(x) → – ∞ as |x| → ∞ Explanation: The negative leading coefficient of the highest even power forces the function downward. 29. Which method is used to factor f(x) = x² – 5x + 6? A. Factorization by grouping B. Quadratic formula C. Synthetic division D. Completing the square Answer: A. Factorization by grouping Explanation: x² – 5x + 6 factors directly into (x – 2)(x – 3).
Answer: A. – 3 Explanation: The coefficient of the highest degree term (x³) is – 3.
36. Factor the polynomial f(x) = x² + 5x + 6. A. (x + 2)(x + 3) B. (x + 1)(x + 6) C. (x + 3)(x + 4) D. (x + 2)(x + 4) Answer: A. (x + 2)(x + 3) Explanation: 2 and 3 add to 5 and multiply to 6. 37. What is the remainder when dividing f(x) = 2x³ + 3x² – x + 5 by x – 1? A. 9 B. 5 C. 2 D. 0 Answer: A. 9 Explanation: By the Remainder Theorem, substitute x = 1: 2 + 3 – 1 + 5 = 9. 38. Determine the zeros of f(x) = x² – 2x – 3. A. x = 3 and x = – 1 B. x = 1 and x = – 3 C. x = 3 only D. x = – 1 only Answer: A. x = 3 and x = – 1 Explanation: Factoring gives (x – 3)(x + 1) = 0. 39. The graph of f(x) = (x – 1)(x + 2) crosses the x-axis at which points? A. x = 1 and x = – 2 B. x = – 1 and x = 2 C. x = 1 and x = 2 D. x = – 1 and x = – 2 Answer: A. x = 1 and x = – 2 Explanation: Setting each factor equal to zero yields the intercepts. 40. Identify the vertical asymptote for f(x) = (x + 3)/(x² – 4). A. x = 2 and x = – 2 B. x = – 3 C. x = 4 D. x = 0 Answer: A. x = 2 and x = – 2 Explanation: Denom. factors as (x – 2)(x + 2); zeros of denominator give vertical asymptotes. 41. What is the domain of f(x) = (x + 2)/(x² – 9)? A. All real numbers except x = 3 and x = – 3 B. All real numbers C. x ≠ 0
D. x > 0 Answer: A. All real numbers except x = 3 and x = – 3 Explanation: Denom. factors as (x – 3)(x + 3); zeros must be excluded.
42. Solve the rational equation (x – 1)/(x + 2) = 2. A. x = – 5 B. x = 5 C. x = – 1 D. x = 1 Answer: A. x = – 5 Explanation: Cross-multiply: x – 1 = 2(x + 2) leads to x – 1 = 2x + 4, so x = – 5. 43. Use synthetic division to divide f(x) = 2x³ + 3x² – x + 4 by x + 1. What is the quotient’s leading term? A. 2x² B. 2x³ C. – 2x² D. x² Answer: A. 2x² Explanation: Synthetic division gives a quotient beginning with 2x². 44. What is the end behavior of f(x) = 5x³ – x? A. f(x) → ∞ as x → ∞ and f(x) → – ∞ as x → – ∞ B. f(x) → – ∞ as x → ∞ C. f(x) is bounded D. f(x) → 0 Answer: A. f(x) → ∞ as x → ∞ and f(x) → – ∞ as x → – ∞ Explanation: A positive odd-degree leading term determines this behavior. 45. Which factor is common in f(x) = x³ – 3x² + 2x? A. x B. x – 1 C. x – 2 D. x² Answer: A. x Explanation: Each term contains x; factoring yields x(x² – 3x + 2). 46. What is the horizontal asymptote for f(x) = (3x + 1)/(x – 2)? A. y = 3 B. y = 1/ C. y = 0 D. There is no horizontal asymptote Answer: D. There is no horizontal asymptote Explanation: The numerator and denominator are first degree, so the horizontal asymptote is y = 3/1 = 3; however, because the degrees are equal, answer A is correct. [Note: Correct answer is A. y = 3] Explanation (revised): For functions where numerator and denominator have the same degree, the horizontal asymptote is the ratio of leading coefficients, 3/1 = 3.
Answer: A. x = 4 Explanation: 16 is 2⁴.
53. What is the logarithmic form of 10³ = 1000? A. log₁₀(1000) = 3 B. log₁₀(3) = 1000 C. log₁₀(1000) = 10³ D. log₁₀(10³) = 1000 Answer: A. log₁₀(1000) = 3 Explanation: Converting exponential form to logarithmic form gives log₁₀(1000) = 3. 54. Solve log₂(8) =? A. 3 B. 2 C. 4 D. 1 Answer: A. 3 Explanation: 2³ = 8. 55. Which logarithmic property is demonstrated by log(ab) = log a + log b? A. Product rule B. Quotient rule C. Power rule D. Change of base rule Answer: A. Product rule Explanation: This is the product property of logarithms. 56. Express log₄(16) in simplest form. A. 2 B. 4 C. 1/ D. 8 Answer: A. 2 Explanation: Since 4² = 16, log₄(16) = 2. 57. Solve the exponential equation 3^(x – 1) = 9. A. x = 3 B. x = 2 C. x = 4 D. x = 1 Answer: A. x = 3 Explanation: 9 = 3², so x – 1 = 2 and x = 3. 58. What is the change of base formula for log_b(a)? A. log_b(a) = log_c(a)/log_c(b) B. log_b(a) = log_b(c)/log_a(c) C. log_b(a) = log(a) + log(b)
D. log_b(a) = log_b(a) – log_b(c) Answer: A. log_b(a) = log_c(a)/log_c(b) Explanation: This is the standard change of base formula.
59. Graphing exponential functions: What does f(x) = 2^(x – 3) represent? A. A horizontal shift of 3 units to the right from f(x) = 2ˣ B. A vertical shift upward by 3 C. A reflection across the y-axis D. A stretch by a factor of 3 Answer: A. A horizontal shift of 3 units to the right Explanation: Replacing x with (x – 3) shifts the graph right. 60. What is the inverse of f(x) = eˣ? A. f⁻¹(x) = ln x B. f⁻¹(x) = e^(–x) C. f⁻¹(x) = 1/eˣ D. f⁻¹(x) = log₁₀ x Answer: A. f⁻¹(x) = ln x Explanation: The natural logarithm is the inverse of the exponential function. 61. Which logarithmic equation is equivalent to ln x = 2? A. e² = x B. 2 = x C. ln 2 = x D. x = 2e Answer: A. e² = x Explanation: Exponentiating both sides gives x = e². 62. Solve for x: ln x = 0. A. x = 1 B. x = 0 C. x = e D. x = – 1 Answer: A. x = 1 Explanation: ln 1 = 0 since e⁰ = 1. 63. Which function models exponential decay? A. f(x) = 3(1/2)ˣ B. f(x) = 3(2)ˣ C. f(x) = 3eˣ D. f(x) = 3ˣ Answer: A. f(x) = 3(1/2)ˣ Explanation: A base between 0 and 1 represents decay. 64. Solve 5ˣ = 125. A. x = 3 B. x = 5
B. log a + log b C. log a – log b D. log(a/b) Answer: A. b log a Explanation: This is the power rule for logarithms.
71. Determine the value of e⁰. A. 1 B. 0 C. e D. – 1 Answer: A. 1 Explanation: Any nonzero number raised to the 0 power equals 1. 72. Solve the equation 2ˣ = 8. A. x = 3 B. x = 2 C. x = 4 D. x = – 3 Answer: A. x = 3 Explanation: 8 = 2³. 73. Express the exponential function f(x) = 3ˣ in logarithmic form. A. log₃ y = x B. log y = 3x C. ln y = x/ D. log₃ x = y Answer: A. log₃ y = x Explanation: Converting 3ˣ = y gives log₃ y = x. 74. What is the inverse of f(x) = ln x? A. f⁻¹(x) = eˣ B. f⁻¹(x) = ln x C. f⁻¹(x) = x D. f⁻¹(x) = 1/x Answer: A. f⁻¹(x) = eˣ Explanation: The exponential function is the inverse of the natural logarithm. 75. If f(x) = 10ˣ, what is f(–1)? A. 0. B. – 0. C. 10 D. – 10 Answer: A. 0. Explanation: 10^(–1) = 1/10 = 0.1.
76. Solve the system: x + y = 5 and x – y = 1. A. x = 3, y = 2 B. x = 2, y = 3 C. x = 4, y = 1 D. x = 1, y = 4 Answer: A. x = 3, y = 2 Explanation: Adding the equations yields 2x = 6, so x = 3; then y = 2. 77. Which method involves solving one equation for a variable and substituting into the other? A. Substitution method B. Elimination method C. Graphical method D. Matrix method Answer: A. Substitution method Explanation: Substitution involves isolating one variable and replacing it in the second equation. 78. Solve: 2x + 3y = 12 and x – y = 2. A. x = 4, y = 2 B. x = 3, y = 3 C. x = 5, y = 1 D. x = 2, y = 0 Answer: A. x = 4, y = 2 Explanation: From x – y = 2, x = y + 2; substitute in first equation to solve. 79. In graphing linear inequalities, which region satisfies y > 2x + 1? A. The region above the line y = 2x + 1 B. The region below the line C. The line itself D. None of the above Answer: A. The region above the line y = 2x + 1 Explanation: y must be greater than the line’s value. 80. Solve the system: x² + y² = 25 and y = 3. A. x = 4 and x = – 4 B. x = 5 C. x = 3 D. x = 0 Answer: A. x = 4 and x = – 4 Explanation: Substitute y = 3 into the circle equation and solve for x. 81. What is the solution for x in the equation x + 2 = 5? A. x = 3 B. x = 2 C. x = 5 D. x = – 3 Answer: A. x = 3 Explanation: Subtract 2 from both sides.
88. Which inequality represents “x is at least 5”? A. x ≥ 5 B. x > 5 C. x ≤ 5 D. x < 5 Answer: A. x ≥ 5 Explanation: “At least” means greater than or equal to. 89. Solve the system: 2x + y = 10 and x – y = 1. A. x = 11/3, y = 10/ B. x = 3, y = 4 C. x = 4, y = 2 D. x = 5, y = 0 Answer: A. x = 11/3, y = 10/ Explanation: Solve by adding and subtracting equations. 90. Which method eliminates a variable by adding the equations? A. Elimination method B. Substitution method C. Graphing method D. Matrix method Answer: A. Elimination method Explanation: Adding or subtracting cancels out one variable. 91. Identify the solution to: 3x + 2y = 12 and 6x + 4y = 24. A. Infinitely many solutions B. No solution C. A unique solution D. x = 0, y = 6 Answer: A. Infinitely many solutions Explanation: The second equation is a multiple of the first. 92. Solve: x – y = 2 and x + y = 8. A. x = 5, y = 3 B. x = 4, y = 2 C. x = 6, y = 4 D. x = 3, y = 1 Answer: A. x = 5, y = 3 Explanation: Adding gives 2x = 10, so x = 5 and y = 3. 93. In linear programming, what is the feasible region? A. The set of all solutions that satisfy all constraints B. The optimal solution only C. The region outside all constraints D. A single point Answer: A. The set of all solutions that satisfy all constraints Explanation: The feasible region is defined by the system of inequalities.
94. Solve the inequality – x + 3 ≥ 5. A. x ≤ – 2 B. x ≥ – 2 C. x ≤ 2 D. x ≥ 2 Answer: A. x ≤ – 2 Explanation: – x ≥ 2, so x ≤ – 2. 95. In a system of inequalities, what does a boundary line represent? A. A line where the inequality changes from true to false B. The solution itself C. A region to be excluded D. None of the above Answer: A. A line where the inequality changes from true to false Explanation: The boundary is where the inequality holds as an equality. 96. Solve the system: x² = 9 and x + y = 5. A. x = 3 yields y = 2 and x = – 3 yields y = 8 B. x = 3 yields y = 8 and x = – 3 yields y = 2 C. x = 0 D. Only x = 3 works Answer: A. x = 3 yields y = 2 and x = – 3 yields y = 8 Explanation: x² = 9 gives x = 3 or – 3; substitute into the second equation. 97. Which method is characterized by adding or subtracting equations to remove a variable? A. Elimination method B. Substitution method C. Graphical method D. Factoring method Answer: A. Elimination method Explanation: This method cancels out one variable by combining equations. 98. Solve: 4x + 5y = 20 and 2x – y = 1. A. x = 3, y = 2. B. x = 2, y = 2. C. x = 3, y = 1 D. x = 2, y = 1 Answer: A. x = 3, y = 2. Explanation: Solve the second equation for y and substitute into the first. 99. Identify the solution of the inequality x/2 ≤ 3. A. x ≤ 6 B. x < 6 C. x ≥ 6 D. x = 6 Answer: A. x ≤ 6 Explanation: Multiply both sides by 2.
106. Determine the determinant of the matrix [[4, 2], [3, 1]]. A. (4×1) – (2×3) = – 2 B. (4×3) – (2×1) = 10 C. (4+1) – (2+3) = 0 D. (4×2) – (3×1) = 5 Answer: A. – 2 Explanation: 4(1) – 2(3) = 4 – 6 = – 2. 107. Which matrix operation is essential for finding the inverse of a matrix? A. Determinant calculation B. Row addition C. Element-wise multiplication D. Transposition only Answer: A. Determinant calculation Explanation: The inverse exists only if the determinant is nonzero. 108. Calculate the inverse of a 2 × 2 matrix [[a, b], [c, d]] (assuming ad – bc ≠ 0). A. (1/(ad – bc))[[d, – b], [–c, a]] B. [[d, – b], [–c, a]] C. (1/(a + d))[[d, – b], [–c, a]] D. [[a, b], [c, d]] Answer: A. (1/(ad – bc))[[d, – b], [–c, a]] Explanation: This is the formula for the inverse of a 2 × 2 matrix. 109. What is the result of subtracting matrix B = [[2, 1], [0, 3]] from A = [[5, 4], [3, 2]]? A. [[3, 3], [3, – 1]] B. [[7, 5], [3, – 1]] C. [[3, 3], [–3, – 1]] D. [[5, 4], [3, 2]] Answer: A. [[5–2, 4–1], [3–0, 2–3]] = [[3, 3], [3, – 1]] Explanation: Subtract corresponding entries. 110. What is Cramer’s rule used for? A. Solving systems of linear equations using determinants B. Finding the inverse of a matrix C. Performing matrix multiplication D. Solving quadratic equations Answer: A. Solving systems of linear equations using determinants Explanation: Cramer’s rule expresses solutions in terms of determinants. 111. Find the determinant of [[2, 5], [1, 3]]. A. (2×3) – (5×1) = 1 B. (2×1) – (5×3) = – 13 C. (2+3) – (5+1) = – 1 D. (2×5) – (1×3) = 7 Answer: A. 1 Explanation: 2(3) – 5(1) = 6 – 5 = 1.
112. Multiply matrices A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]]. A. [[2, 1], [4, 3]] B. [[1, 2], [3, 4]] C. [[2, 0], [4, 0]] D. [[0, 3], [1, 4]] Answer: A. [[(1×0+2×1), (1×1+2×0)], [(3×0+4×1), (3×1+4×0)]] = [[2, 1], [4, 3]] Explanation: Multiply as per matrix multiplication rules. 113. What is the identity matrix of order 2? A. [[1, 0], [0, 1]] B. [[0, 1], [1, 0]] C. [[1, 1], [1, 1]] D. [[2, 0], [0, 2]] Answer: A. [[1, 0], [0, 1]] Explanation: The identity matrix has ones on the diagonal and zeros elsewhere. 114. Which operation is not defined for matrices of different dimensions? A. Addition B. Scalar multiplication C. Determinant calculation D. Transposition Answer: A. Addition Explanation: Matrices must be the same size to be added. 115. What is the trace of matrix [[2, 3], [4, 5]]? A. 7 B. 8 C. 9 D. 5 Answer: A. 2 + 5 = 7 Explanation: The trace is the sum of the diagonal elements. 116. Determine the determinant of the 3 × 3 matrix [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. A. 1(1×0 – 4×6) – 2(0×0 – 4×5) + 3(0×6 – 1×5) = 1(–24) – 2(–20) + 3(–5) = – 24 + 40 – 15 = 1 B. 0 C. 10 D. – 1 Answer: A. 1 Explanation: Using cofactor expansion yields a determinant of 1. 117. For a square matrix, when is an inverse defined? A. When the determinant is nonzero B. Always C. When the matrix is symmetric D. Only for diagonal matrices Answer: A. When the determinant is nonzero Explanation: A zero determinant indicates a singular matrix.
