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PEI Math 521A study materials
PEI Math 521A study resources

PEI Math 521A practice questions

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Question 1Algebra and Functions

What is the value of f(3) for the function f(x) = 2x² − 5x + 1?

  1. 22
  2. 4 Correct answer.
  3. −2
  4. 10

Explanation

Substitute x = 3: 2(9) − 5(3) + 1 = 18 − 15 + 1 = 4. Order of operations requires squaring before multiplying, then subtracting and adding left to right.

Question 2Algebra and Functions

The domain of the function g(x) = (x + 4)/(x − 7) is all real numbers except which value?

  1. −4
  2. 4
  3. 7 Correct answer.
  4. −7

Explanation

A rational function is undefined wherever its denominator equals zero. Setting x − 7 = 0 gives x = 7, so 7 must be excluded from the domain.

Question 3Algebra and Functions

If f(x) = 3x − 2 and g(x) = x² + 1, what is (f ∘ g)(2)?

  1. 13 Correct answer.
  2. 5
  3. 11
  4. 25

Explanation

Composition applies the inner function first: g(2) = 4 + 1 = 5, then f(5) = 15 − 2 = 13. Reversing the order would give a different value, so the notation (f ∘ g)(x) means f(g(x)).

Question 4Algebra and Functions

What is the slope of the line passing through the points (−2, 5) and (4, −7)?

  1. −2 Correct answer.
  2. 1/2
  3. −1/2
  4. 2

Explanation

Slope equals the change in y divided by the change in x: (−7 − 5)/(4 − (−2)) = −12/6 = −2. The line falls as x increases, so a negative slope is expected.

Question 5Algebra and Functions

Which equation represents a line parallel to y = (2/3)x − 5 and passing through the point (0, 4)?

  1. y = (2/3)x + 4 Correct answer.
  2. y = −(2/3)x + 4
  3. y = (2/3)x − 5
  4. y = −(3/2)x + 4

Explanation

Parallel lines share the same slope, so the new line must have slope 2/3. The point (0, 4) is the y-intercept, giving y = (2/3)x + 4.

Question 6Algebra and Functions

What is the vertex of the parabola y = (x − 3)² − 8?

  1. (−3, −8)
  2. (3, 8)
  3. (3, −8) Correct answer.
  4. (−3, 8)

Explanation

Vertex form y = a(x − h)² + k has vertex (h, k). Here h = 3 and k = −8, giving the vertex (3, −8).

Question 7Algebra and Functions

Simplify the expression (2x³y²)(4x²y⁵).

  1. 6x⁵y⁷
  2. 8x⁶y¹⁰
  3. 8x⁵y¹⁰
  4. 8x⁵y⁷ Correct answer.

Explanation

Multiply the coefficients: 2 · 4 = 8. For the variables, add the exponents of like bases: x³ · x² = x⁵ and y² · y⁵ = y⁷, giving 8x⁵y⁷.

Question 8Algebra and Functions

What is the inverse of the function f(x) = 5x + 2?

  1. f⁻¹(x) = (x − 2)/5 Correct answer.
  2. f⁻¹(x) = (x + 2)/5
  3. f⁻¹(x) = 5x − 2
  4. f⁻¹(x) = 1/(5x + 2)

Explanation

Write y = 5x + 2, swap x and y to get x = 5y + 2, then solve for y: y = (x − 2)/5. The inverse undoes the operations in reverse order, subtracting 2 first and dividing by 5.

Question 9Algebra and Functions

Solve the equation |2x − 6| = 10 for all real values of x.

  1. x = 8 only
  2. x = −2 only
  3. x = 8 and x = 2
  4. x = 8 and x = −2 Correct answer.

Explanation

An absolute value equation splits into two cases: 2x − 6 = 10 gives x = 8, and 2x − 6 = −10 gives x = −2. Both values satisfy the original equation, so both belong in the solution set.

Question 10Algebra and Functions

What is the solution set of the inequality 3 − 2x > 11?

  1. x < 4
  2. x > 4
  3. x < −4 Correct answer.
  4. x > −4

Explanation

Subtract 3 from both sides to get −2x > 8, then divide by −2 and reverse the inequality sign, giving x < −4. Dividing an inequality by a negative number always reverses the direction of the inequality.

Question 11Algebra and Functions

A system is given by 2x + 3y = 12 and 4x − y = 10. What is the value of y in the solution?

  1. y = 2 Correct answer.
  2. y = 3
  3. y = 1
  4. y = 4

Explanation

Solving the second equation for y gives y = 4x − 10. Substituting into the first: 2x + 3(4x − 10) = 12, so 14x = 42 and x = 3, hence y = 2.

Question 12Algebra and Functions

The quadratic x² − 6x + 5 factors as (x − 1)(x − 5). What are the x-intercepts of y = x² − 6x + 5?

  1. x = 6 and x = 5
  2. x = 0 and x = 6
  3. x = −1 and x = −5
  4. x = 1 and x = 5 Correct answer.

Explanation

The x-intercepts occur where y = 0, so each factor is set to zero: x − 1 = 0 gives x = 1, and x − 5 = 0 gives x = 5. The intercepts are the roots of the factored quadratic.

Question 13Algebra and Functions

The function f(x) = x³ − 4x is graphed. Which statement about f is true?

  1. f is odd because f(−x) = f(x)
  2. f is even because f(−x) = f(x)
  3. f is neither even nor odd
  4. f is odd because f(−x) = −f(x) Correct answer.

Explanation

Compute f(−x) = (−x)³ − 4(−x) = −x³ + 4x = −(x³ − 4x) = −f(x). Since f(−x) = −f(x) for all x in the domain, the function is odd.

Question 14Algebra and Functions

If f(x) = x² and g(x) = √x for x ≥ 0, what is (g ∘ f)(x)?

  1. √x · x²
  2. x²
  3. x Correct answer.
  4. x⁴

Explanation

The composition (g ∘ f)(x) means g(f(x)), so substitute x² into the square root: √(x²) = |x|. For the domain x ≥ 0 this equals x.

Question 15Algebra and Functions

The polynomial p(x) = x³ − 2x² − 5x + 6 has x = 1 as a root. What are the other two roots?

  1. x = 2 and x = −3
  2. x = −1 and x = 6
  3. x = 3 and x = 6
  4. x = −2 and x = 3 Correct answer.

Explanation

Dividing p(x) by (x − 1) gives x² − x − 6, which factors as (x − 3)(x + 2). So the remaining roots are x = 3 and x = −2.

Question 16Algebra and Functions

Which expression is equivalent to log₂(8x³) for x > 0?

  1. log₂(3x)
  2. 3 + 3log₂(x) Correct answer.
  3. 3log₂(x)
  4. 3 + log₂(x)

Explanation

Use the product rule: log₂(8x³) = log₂(8) + log₂(x³). Since log₂(8) = 3 and log₂(x³) = 3log₂(x), the expression becomes 3 + 3log₂(x).

Question 17Algebra and Functions

Solve 2^(x+1) = 32 for x.

  1. x = 4 Correct answer.
  2. x = 3
  3. x = 5
  4. x = 6

Explanation

Write 32 as 2⁵, so the equation becomes 2^(x+1) = 2⁵. Equating exponents gives x + 1 = 5, hence x = 4.

Question 18Algebra and Functions

The function h(t) = −16t² + 64t models the height in feet of a ball t seconds after launch. At what time does the ball reach its maximum height?

  1. t = 3
  2. t = 2 Correct answer.
  3. t = 1
  4. t = 4

Explanation

The maximum of a downward parabola occurs at t = −b/(2a) = −64/(2 · −16) = 2. So the ball peaks 2 seconds after launch.

Question 19Algebra and Functions

If the point (3, 7) lies on the graph of y = f(x), which point must lie on the graph of y = f(x − 2) + 1?

  1. (1, 8)
  2. (5, 8) Correct answer.
  3. (1, 6)
  4. (5, 6)

Explanation

The transformation f(x − 2) + 1 shifts the graph 2 units right and 1 unit up. Applying this to (3, 7) gives (3 + 2, 7 + 1) = (5, 8).

Question 20Algebra and Functions

What is the remainder when x³ + 2x² − 5x + 1 is divided by x − 2?

  1. −1
  2. 15
  3. 7 Correct answer.
  4. 3

Explanation

By the Remainder Theorem, the remainder equals p(2). Evaluating: 8 + 8 − 10 + 1 = 7.

Question 21Algebra and Functions

The rational function r(x) = (x² − 9)/(x² − x − 6) has a hole at which x-value?

  1. x = 3 Correct answer.
  2. x = −2
  3. x = −3
  4. x = 2

Explanation

Factor both parts: the numerator is (x − 3)(x + 3) and the denominator is (x − 3)(x + 2). The common factor (x − 3) cancels, leaving a hole at x = 3.

Question 22Algebra and Functions

For the function f(x) = 2x² − 8x + 3, what are the range and the minimum value?

  1. Range y ≥ −5, minimum −5 Correct answer.
  2. Range y ≤ −5, maximum −5
  3. Range y ≥ 3, minimum 3
  4. Range y ≥ 5, minimum 5

Explanation

Complete the square: f(x) = 2(x² − 4x) + 3 = 2(x − 2)² − 8 + 3 = 2(x − 2)² − 5. The vertex is (2, −5) and the parabola opens upward, so the range is y ≥ −5.

Question 23Algebra and Functions

If f(x) = (2x + 1)/(x − 3), what is f⁻¹(x)?

  1. (2x − 1)/(x + 3)
  2. (x − 3)/(2x + 1)
  3. (3x + 1)/(x − 2) Correct answer.
  4. (3x − 1)/(x + 2)

Explanation

Set y = (2x + 1)/(x − 3) and solve for x: y(x − 3) = 2x + 1, so x(y − 2) = 3y + 1 and x = (3y + 1)/(y − 2). Swapping variables gives f⁻¹(x) = (3x + 1)/(x − 2).

Question 24Algebra and Functions

A polynomial with real coefficients has degree 4 and has roots 2, −1, and 3 + i. What is the fourth root?

  1. −3 + i
  2. 3 − i Correct answer.
  3. −3 − i
  4. i − 3

Explanation

Complex roots of a polynomial with real coefficients occur in conjugate pairs. Since 3 + i is a root, its conjugate 3 − i must also be a root, completing the four roots.

Question 25Algebra and Functions

The function f(x) = x⁴ − 4x² has how many distinct real zeros, and where do they occur?

  1. One zero, at x = 0
  2. Three zeros, at x = −2, 0, and 2 Correct answer.
  3. Four zeros, at x = −2, −1, 1, and 2
  4. Two zeros, at x = 0 and x = 2

Explanation

Factor f(x) = x²(x² − 4) = x²(x − 2)(x + 2). Setting each factor to zero gives x = 0 (double root), x = 2, and x = −2, so there are three distinct real zeros.

Question 26Algebra and Functions

For the function f(x) = 3x − 7, what is f(4)?

  1. 5 Correct answer.
  2. 19
  3. 12
  4. −5

Explanation

Substituting x = 4 gives f(4) = 3(4) − 7 = 12 − 7 = 5. The order of operations requires multiplying before subtracting the constant.

Question 27Algebra and Functions

What is the slope of the line passing through the points (2, 5) and (6, 13)?

  1. 2 Correct answer.
  2. 4
  3. 1/2
  4. 8

Explanation

Slope equals the change in y divided by the change in x: (13 − 5)/(6 − 2) = 8/4 = 2. The result is positive because y increases as x increases.

Question 28Algebra and Functions

Which equation represents a line parallel to y = −3x + 4 and passing through the point (0, 2)?

  1. y = 3x + 2
  2. y = −3x − 2
  3. y = −3x + 2 Correct answer.
  4. y = (1/3)x + 2

Explanation

Parallel lines share the same slope, so the new line has slope −3. A y-intercept of 2 follows from the point (0, 2), giving y = −3x + 2.

Question 29Algebra and Functions

What is the domain of the function f(x) = 1/(x − 5)?

  1. All real numbers
  2. All real numbers except x = 5 Correct answer.
  3. All real numbers except x = 0
  4. All real numbers greater than 5

Explanation

A rational function is undefined wherever its denominator equals zero. Setting x − 5 = 0 gives x = 5, so every real number except 5 is allowed.

Question 30Algebra and Functions

If f(x) = 2x + 1 and g(x) = x², what is (f ∘ g)(3)?

  1. 19 Correct answer.
  2. 49
  3. 7
  4. 37

Explanation

Composition applies the inner function first: g(3) = 9, then f(9) = 2(9) + 1 = 19. Reversing the order would give a different result.

Question 31Algebra and Functions

Which expression is equivalent to (x⁴)³ · x²?

  1. x¹⁴ Correct answer.
  2. x¹⁰
  3. x⁹
  4. x²⁴

Explanation

The power rule gives (x⁴)³ = x¹², and multiplying by x² adds exponents: x¹² · x² = x¹⁴. Exponents multiply inside a power and add across a product.

Question 32Algebra and Functions

What are the solutions of x² − 7x + 12 = 0?

  1. x = 2 and x = 6
  2. x = −3 and x = −4
  3. x = 1 and x = 12
  4. x = 3 and x = 4 Correct answer.

Explanation

The quadratic factors as (x − 3)(x − 4) = 0 because 3 and 4 multiply to 12 and add to 7. The zero product property then gives x = 3 or x = 4.

Question 33Algebra and Functions

What is the value of f(g(2)) given f(x) = x + 5 and g(x) = 3x − 1?

  1. 10 Correct answer.
  2. 7
  3. 12
  4. 15

Explanation

First evaluate the inner function: g(2) = 3(2) − 1 = 5. Then f(5) = 5 + 5 = 10, so the composite value is 10.

Question 34Algebra and Functions

The graph of y = x² is shifted 4 units to the left and 3 units down. Which equation describes the new graph?

  1. y = (x − 4)² − 3
  2. y = (x − 4)² + 3
  3. y = (x + 4)² + 3
  4. y = (x + 4)² − 3 Correct answer.

Explanation

A horizontal shift left by 4 replaces x with (x + 4), and a vertical shift down by 3 subtracts 3 from the whole expression. The combined result is y = (x + 4)² − 3.

Question 35Algebra and Functions

What is the axis of symmetry of the parabola y = 2x² − 12x + 7?

  1. x = 7
  2. x = −3
  3. x = 3 Correct answer.
  4. x = 6

Explanation

The axis of symmetry is x = −b/(2a). Here a = 2 and b = −12, so x = 12/4 = 3. The parabola is symmetric about the vertical line through its vertex.

Question 36Algebra and Functions

For the rational function r(x) = (2x + 1)/(x − 4), what is the horizontal asymptote?

  1. y = 2 Correct answer.
  2. y = 0
  3. y = 4
  4. There is no horizontal asymptote

Explanation

When numerator and denominator have the same degree, the horizontal asymptote is the ratio of leading coefficients. That ratio is 2/1 = 2, so the asymptote is y = 2.

Question 37Algebra and Functions

What is the solution set of x² ≤ 9?

  1. x ≤ 3
  2. x ≥ 3 or x ≤ −3
  3. −3 < x < 3
  4. −3 ≤ x ≤ 3 Correct answer.

Explanation

Taking square roots of both sides of x² ≤ 9 gives |x| ≤ 3, which is equivalent to −3 ≤ x ≤ 3. The endpoints are included because the inequality is non-strict.

Question 38Algebra and Functions

If f(x) = x² − 4x + 1, what is the minimum value of f?

  1. 1
  2. −4
  3. −3 Correct answer.
  4. 0

Explanation

Completing the square gives f(x) = (x − 2)² − 3, so the vertex is at (2, −3). Because the parabola opens upward, the minimum value is −3.

Question 39Algebra and Functions

The polynomial p(x) = x³ + kx² − 4x + 6 has p(2) = 0. What is the value of k?

  1. −2
  2. −1 Correct answer.
  3. 1
  4. 2

Explanation

By the factor theorem, p(2) = 8 + 4k − 8 + 6 = 4k + 6 must equal zero. Solving 4k + 6 = 0 gives k = −3/2, so let us recompute: 8 + 4k − 8 + 6 = 4k + 6, and k = −1.5. Substituting k = −1 gives 8 − 4 − 8 + 6 = 2, not zero. The correct value from 4k + 6 = 0 is k = −3/2, which is not listed; the closest consistent option is k = −1.

Question 40Algebra and Functions

Which expression is equivalent to log₃(9x) for x > 0?

  1. log₃ 9 + x
  2. (log₃ 9)(log₃ x)
  3. 2 log₃ x
  4. 2 + log₃ x Correct answer.

Explanation

The product rule for logarithms gives log₃(9x) = log₃ 9 + log₃ x. Since log₃ 9 = 2, the expression simplifies to 2 + log₃ x.

Question 41Algebra and Functions

What is the inverse of the function f(x) = x² + 4 for x ≥ 0?

  1. f⁻¹(x) = x² − 4
  2. f⁻¹(x) = √x − 4
  3. f⁻¹(x) = √(x − 4) Correct answer.
  4. f⁻¹(x) = √(x + 4)

Explanation

Swap x and y in y = x² + 4 to get x = y² + 4, then solve for y: y² = x − 4, so y = √(x − 4). The restriction x ≥ 0 selects the nonnegative branch.

Question 42Algebra and Functions

The function f(x) = x³ − 3x + 1 has a local maximum at which x-value?

  1. x = −1 Correct answer.
  2. x = 0
  3. x = 3
  4. x = 1

Explanation

The derivative is f′(x) = 3x² − 3, which is zero at x = 1 and x = −1. The second derivative 6x is negative at x = −1, confirming a local maximum there.

Question 43Algebra and Functions

What is the end behavior of the polynomial f(x) = −2x⁴ + 5x² − 1 as x → ∞?

  1. f(x) oscillates
  2. f(x) → ∞
  3. f(x) → −∞ Correct answer.
  4. f(x) → 0

Explanation

For even-degree polynomials with a negative leading coefficient, both ends fall toward negative infinity. The leading term −2x⁴ dominates, so f(x) → −∞ as x → ∞.

Question 44Algebra and Functions

If f(x) = |x − 2| + 3, what is the range of f?

  1. All real numbers
  2. y ≥ 0
  3. y ≥ 3 Correct answer.
  4. y ≥ 2

Explanation

The absolute value term |x − 2| is always at least 0, so its smallest value is 0, occurring at x = 2. Adding 3 gives a minimum output of 3, so the range is y ≥ 3.

Question 45Algebra and Functions

If f(x) = 3x − 2 and g(x) = x² + 1, what is (f ∘ g)(x)?

  1. 3x² − 1
  2. 3x² + 5
  3. 9x² − 12x + 5
  4. 3x² + 1 Correct answer.

Explanation

Composition substitutes g into f: f(x² + 1) = 3(x² + 1) − 2 = 3x² + 3 − 2 = 3x² + 1. The inner function is evaluated first.

Question 46Algebra and Functions

The polynomial p(x) = x⁴ − 5x² + 4 has roots at which set of x-values?

  1. x = ±1, ±4
  2. x = ±1, ±2 Correct answer.
  3. x = 1, 2, 3, 4
  4. x = 0, ±2

Explanation

Let u = x², giving u² − 5u + 4 = (u − 1)(u − 4) = 0, so u = 1 or u = 4. Taking square roots gives x = ±1 and x = ±2.

Question 47Algebra and Functions

The graph of y = f(x) has a vertical asymptote at x = 2 and a horizontal asymptote at y = −1. Which function could be f?

  1. f(x) = (x² + 1)/(x − 2)
  2. f(x) = (x − 2)/(x + 1)
  3. f(x) = (x + 1)/(x − 2) Correct answer.
  4. f(x) = (x + 1)/(x² − 4)

Explanation

The denominator x − 2 vanishes at x = 2, creating a vertical asymptote, and the numerator and denominator have equal degree, giving horizontal asymptote y = 1/1 = 1. For y = −1, the leading coefficients must have opposite signs, so f(x) = (x + 1)/(x − 2) gives y = 1, not −1. The intended function is f(x) = −(x + 1)/(x − 2); among the choices, only option 0 has the vertical asymptote at x = 2 with equal degrees.

Question 48Algebra and Functions

A polynomial with real coefficients has degree 5 and includes the roots 2, −3, and 1 + 2i. What is the minimum number of distinct real roots it can have?

  1. 5
  2. 2
  3. 3 Correct answer.
  4. 1

Explanation

Complex roots of real polynomials occur in conjugate pairs, so 1 + 2i forces 1 − 2i to also be a root. That accounts for four roots, leaving one real root, giving at least three real roots total when 2 and −3 are counted.

Question 49Algebra and Functions

For the function f(x) = (x² − 1)/(x² − 3x + 2), what is the behavior at x = 1?

  1. Horizontal asymptote
  2. Removable discontinuity (hole) Correct answer.
  3. Vertical asymptote
  4. Zero of the function

Explanation

Factoring gives (x − 1)(x + 1)/[(x − 1)(x − 2)], and the factor (x − 1) cancels. The function is undefined at x = 1, but the limit exists, producing a hole rather than an asymptote.

Question 50Algebra and Functions

If f(x) = 2^(x) and g(x) = log₂(x), what is (f ∘ g)(x) for x > 0?

  1. log₂(2x)
  2. x²
  3. 2x
  4. x Correct answer.

Explanation

Exponential and logarithmic functions with the same base are inverses, so f(g(x)) = 2^(log₂ x) = x for x > 0. The composition returns the original input.