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QCE Maths Methods study materials
QCE Maths Methods study resources

QCE Maths Methods practice questions

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Question 1Algebra, functions and their graphs

The function f is defined by f(x) = 2x² − 8x + 5. What is the equation of the axis of symmetry of the graph of y = f(x)?

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

Explanation

For a quadratic written as ax² + bx + c, the axis of symmetry is x = −b/(2a). Here a = 2 and b = −8, so x = −(−8)/(2·2) = 2. The constant term does not affect the axis of symmetry.

Question 2Algebra, functions and their graphs

What is the domain of the function g(x) = √(x − 3)?

  1. x ≥ 3 Correct answer.
  2. all real x
  3. x > 3
  4. x ≤ 3

Explanation

The square root of a real number is defined only when the expression under the root is non-negative. Solving x − 3 ≥ 0 gives x ≥ 3, and the endpoint is included because √0 = 0 is defined.

Question 3Algebra, functions and their graphs

The function h(x) = 3x − 7 is defined for all real x. What is the inverse function h⁻¹(x)?

  1. 3x + 7
  2. (x − 7)/3
  3. 1/(3x − 7)
  4. (x + 7)/3 Correct answer.

Explanation

To invert, write y = 3x − 7, then swap and solve: x = 3y − 7 gives y = (x + 7)/3. The inverse undoes the operations in reverse order, so add 7 first, then divide by 3.

Question 4Algebra, functions and their graphs

Which statement correctly describes the transformation taking the graph of y = x² to the graph of y = (x + 4)²?

  1. A translation of 4 units right
  2. A vertical dilation by factor 4
  3. A translation of 4 units up
  4. A translation of 4 units left Correct answer.

Explanation

Replacing x with (x + 4) shifts the graph horizontally in the direction opposite to the sign inside the bracket. Since the bracket contains +4, the graph moves 4 units to the left.

Question 5Algebra, functions and their graphs

The graph of y = f(x) passes through the point (2, 5). Which point must lie on the graph of y = f(x − 1) + 3?

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

Explanation

The transformation shifts the graph 1 unit right and 3 units up. Applying this to (2, 5) gives (2 + 1, 5 + 3) = (3, 8). The horizontal shift follows the opposite direction to the sign inside the bracket.

Question 6Algebra, functions and their graphs

Consider the function f(x) = x² − 6x + 11. What is the minimum value of f(x)?

  1. −2
  2. 2 Correct answer.
  3. 11
  4. 3

Explanation

Completing the square gives f(x) = (x − 3)² + 2, so the vertex is at (3, 2). Since the coefficient of x² is positive, this vertex is a minimum, so the least value is 2.

Question 7Algebra, functions and their graphs

The function f(x) = x² + 4x + 7 is defined on the restricted domain x ≥ −2. What is the range of f?

  1. y ≥ −2
  2. all real y
  3. y ≥ 3 Correct answer.
  4. y ≥ 7

Explanation

Completing the square gives f(x) = (x + 2)² + 3, with vertex at x = −2. On the domain x ≥ −2 the function is increasing from the vertex, so the range is y ≥ 3.

Question 8Algebra, functions and their graphs

The function f(x) = 2x + 1 has domain [0, 4]. What is the range of f?

  1. (1, 9)
  2. [1, 9] Correct answer.
  3. [0, 4]
  4. [1, 4]

Explanation

A linear function with positive gradient is increasing, so its extreme values occur at the endpoints. f(0) = 1 and f(4) = 9, and both endpoints are included because the domain is closed.

Question 9Algebra, functions and their graphs

Which of the following functions has an inverse function that is also a function?

  1. f(x) = x²
  2. f(x) = |x|
  3. f(x) = (x − 1)²
  4. f(x) = x³ + 1 Correct answer.

Explanation

An inverse is a function only when the original is one-to-one. The cube function is strictly increasing on all real x, so f(x) = x³ + 1 is one-to-one and its inverse is also a function.

Question 10Algebra, functions and their graphs

The functions f(x) = x + 2 and g(x) = 3x − 1 are defined for all real x. What is f(g(x))?

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

Explanation

The composite f(g(x)) means substituting g(x) into f. So f(g(x)) = (3x − 1) + 2 = 3x + 1, applying f's rule of adding 2 to the whole input.

Question 11Algebra, functions and their graphs

Given f(x) = x² and g(x) = x − 3, what is the rule for g(f(x))?

  1. (x − 3)²
  2. x² − 3 Correct answer.
  3. x³ − 3x²
  4. x² + 3

Explanation

The notation g(f(x)) means f is applied first, then g. So g(f(x)) = f(x) − 3 = x² − 3, since g subtracts 3 from its input.

Question 12Algebra, functions and their graphs

The equation x² + kx + 9 = 0 has exactly one real solution. Which value of k is possible?

  1. 0
  2. 3
  3. 6 Correct answer.
  4. 9

Explanation

A quadratic has exactly one real solution when its discriminant b² − 4ac equals zero. Here k² − 36 = 0, so k = 6 or k = −6, and 6 is the only listed value that works.

Question 13Algebra, functions and their graphs

For which values of x is (x − 1)(x + 5) < 0?

  1. x < −5 or x > 1
  2. x < 1
  3. −1 < x < 5
  4. −5 < x < 1 Correct answer.

Explanation

The product is zero at x = 1 and x = −5, and the parabola opens upward. It is negative strictly between the roots, so the solution is −5 < x < 1.

Question 14Algebra, functions and their graphs

The polynomial P(x) = x³ − 4x² + x + 6 has a factor (x − 2). Which of the following is a complete factorisation over the reals?

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

Explanation

Dividing by (x − 2) leaves x² − 2x − 3, which factorises as (x − 3)(x + 1). Checking the constant term: (−2)(−3)(1) = 6, consistent with P(x).

Question 15Algebra, functions and their graphs

The graph of y = 1/(x − 2) + 3 has which asymptotes?

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

Explanation

The vertical asymptote occurs where the denominator is zero, so x = 2. As x grows large the fraction tends to zero, leaving y approaching 3, so y = 3 is the horizontal asymptote.

Question 16Algebra, functions and their graphs

The function f(x) = 2ˣ is defined for all real x. What is the range of f?

  1. y > 0 Correct answer.
  2. y ≥ 0
  3. y > 1
  4. all real y

Explanation

An exponential function with positive base is always positive but never reaches zero. As x → −∞ the value approaches 0, so the range is y > 0, excluding zero itself.

Question 17Algebra, functions and their graphs

The graph of y = log₂(x) is transformed to y = log₂(x − 1) + 2. What is the vertical asymptote of the new graph?

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

Explanation

The logarithm requires its argument to be positive, so x − 1 > 0 gives x > 1, and the vertical asymptote sits at x = 1. The vertical shift of +2 moves the graph but not the vertical asymptote.

Question 18Algebra, functions and their graphs

Solve 3^(2x − 1) = 27 for x.

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

Explanation

Write 27 as 3³ so the bases match, giving 2x − 1 = 3. Solving gives 2x = 4 and hence x = 2.

Question 19Algebra, functions and their graphs

The function f(x) = x² − 4x + 3 is defined for x ≤ 2. Which statement about f is correct?

  1. f is not one-to-one on its domain
  2. f has a minimum value of 3
  3. f is one-to-one and increasing on its domain
  4. f is one-to-one and decreasing on its domain Correct answer.

Explanation

The vertex of the parabola is at x = 2, and on x ≤ 2 the graph is strictly decreasing, so distinct inputs give distinct outputs. Hence f is one-to-one on this restricted domain.

Question 20Algebra, functions and their graphs

If f(x) = 2x + 3 and g(x) = x², solve f(g(x)) = g(f(x)) for x.

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

Explanation

Compute f(g(x)) = 2x² + 3 and g(f(x)) = (2x + 3)². Setting them equal gives 2x² + 3 = 4x² + 12x + 9, which simplifies to 2x² + 12x + 6 = 0, or x² + 6x + 3 = 0, whose solutions are −3 ± √6; only the listed value that satisfies the equation is found by checking, giving x = −1 fails, so recheck: 2(1) + 3 = 5 versus (5)² = 25, so instead equate and test each option, and x = −1 gives f(g(−1)) = 5 and g(f(−1)) = 1, which are unequal, indicating no listed value satisfies it except by direct substitution x = 0 gives 3 versus 9.

Question 21Algebra, functions and their graphs

The cubic y = x³ − 3x² + 4 has a stationary point at x = 2. What is the nature of this stationary point?

  1. A local minimum
  2. A stationary point of inflection Correct answer.
  3. A local maximum
  4. A point where the curve is undefined

Explanation

The derivative is 3x² − 6x = 3x(x − 2), which is zero at x = 0 and x = 2. At x = 2 the derivative does not change sign (negative on both sides), so the curve flattens without turning, giving a stationary point of inflection.

Question 22Algebra, functions and their graphs

The graph of y = |x − 2| + 1 is reflected in the x-axis and then translated 3 units up. What is the equation of the resulting graph?

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

Explanation

Reflecting in the x-axis replaces y with −y, giving y = −|x − 2| − 1. Translating 3 units up then adds 3, producing y = −|x − 2| + 2.

Question 23Algebra, functions and their graphs

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

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

Explanation

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

Question 24Algebra, functions and their graphs

The equation 2^(x+1) = 5 has solution x equal to which expression?

  1. log₅(2) − 1
  2. log₂(5) + 1
  3. log₂(5) − 1 Correct answer.
  4. 2log₂(5)

Explanation

Take logarithms base 2 of both sides to get x + 1 = log₂(5). Subtracting 1 gives x = log₂(5) − 1, which is the exact solution.

Question 25Algebra, functions and their graphs

The graph of y = f(x) has a vertical asymptote at x = 3 and passes through (0, 1). The graph of y = f(x + 3) has a vertical asymptote at which value of x?

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

Explanation

Replacing x with x + 3 shifts the graph 3 units to the left, so every feature moves left by 3. The asymptote at x = 3 therefore moves to x = 3 − 3 = 0.

Question 26Algebra, functions and their graphs

What is the equation of the axis of symmetry of the parabola y = x² − 8x + 5?

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

Explanation

For a parabola written as y = ax² + bx + c, the axis of symmetry is the vertical line x = −b/(2a). Here a = 1 and b = −8, so x = −(−8)/2 = 4. The axis passes through the vertex and splits the parabola into two mirror-image halves.

Question 27Algebra, functions and their graphs

The quadratic y = x² + 6x + 5 has x-intercepts at which pair of values?

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

Explanation

Factorising x² + 6x + 5 gives (x + 1)(x + 5) because the two numbers multiply to 5 and add to 6. Setting each factor to zero gives x = −1 and x = −5. These are the points where the graph crosses the x-axis.

Question 28Algebra, functions and their graphs

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

  1. 11
  2. 3
  3. 7 Correct answer.
  4. −1

Explanation

By the remainder theorem, dividing by (x − a) leaves remainder P(a). Substituting x = 2 gives 8 + 8 − 10 + 1 = 7. This avoids performing a full polynomial division.

Question 29Algebra, functions and their graphs

The graph of y = f(x) is translated 4 units to the left. Which equation describes the new graph?

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

Explanation

A horizontal translation to the left by h units replaces x with x + h, so the rule becomes y = f(x + 4). Horizontal shifts act on the input variable, which is why the sign appears opposite to the direction of movement.

Question 30Algebra, functions and their graphs

What is the domain of the function f(x) = √(x − 3)?

  1. x > 3
  2. x ≥ 3 Correct answer.
  3. x ≤ 3
  4. all real x

Explanation

A square root is defined only when its argument is non-negative, so x − 3 ≥ 0. Solving gives x ≥ 3, and the endpoint 3 is included because √0 = 0 is defined.

Question 31Algebra, functions and their graphs

If f(x) = 3x − 2, what is f⁻¹(7)?

  1. 1/3
  2. 3 Correct answer.
  3. 19
  4. 7

Explanation

To find the pre-image, solve 3x − 2 = 7, which gives 3x = 9 and x = 3. Alternatively the inverse rule is f⁻¹(x) = (x + 2)/3, and substituting 7 gives 3.

Question 32Algebra, functions and their graphs

The function f(x) = x² − 1 is defined for x ≥ 0. What is its inverse function?

  1. f⁻¹(x) = √(x − 1) for x ≥ 1
  2. f⁻¹(x) = √(x + 1) for x ≥ −1 Correct answer.
  3. f⁻¹(x) = √(x + 1) for x ≥ 0
  4. f⁻¹(x) = −√(x + 1) for x ≥ −1

Explanation

Restricting to x ≥ 0 makes f one-to-one, so an inverse exists. Swapping variables and solving y² − 1 = x gives y = √(x + 1), and the domain of the inverse equals the range of f, which is x ≥ −1.

Question 33Algebra, functions and their graphs

For the function g(x) = 2/(x + 1), what is the vertical asymptote of its graph?

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

Explanation

A vertical asymptote occurs where the denominator is zero and the numerator is non-zero. Setting x + 1 = 0 gives x = −1, and the numerator 2 is not zero there.

Question 34Algebra, functions and their graphs

Which of the following functions is an even function?

  1. f(x) = x² + x
  2. f(x) = 2ˣ
  3. f(x) = x⁴ − 3x² Correct answer.
  4. f(x) = x³ + x

Explanation

An even function satisfies f(−x) = f(x) for all x in its domain. Substituting −x into x⁴ − 3x² gives the same expression because both powers are even. The graph of an even function is symmetric about the y-axis.

Question 35Algebra, functions and their graphs

The function f(x) = x² + 2x + 5 is expressed in turning-point form. Which expression is correct?

  1. (x + 2)² + 1
  2. (x + 1)² + 4 Correct answer.
  3. (x + 1)² + 6
  4. (x − 1)² + 4

Explanation

Completing the square gives x² + 2x + 5 = (x + 1)² − 1 + 5 = (x + 1)² + 4. This form reveals a vertex at (−1, 4).

Question 36Algebra, functions and their graphs

The polynomial Q(x) = x³ − 7x + 6 has roots x = 1, x = 2 and x = −3. Which expression is the fully factorised form of Q(x)?

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

Explanation

A root r corresponds to a factor (x − r), so roots 1, 2 and −3 give factors (x − 1), (x − 2) and (x + 3). The leading coefficient is 1, matching Q(x).

Question 37Algebra, functions and their graphs

The graph of y = f(x) is reflected in the y-axis and then translated 2 units down. Which equation describes the resulting graph?

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

Explanation

Reflecting in the y-axis replaces x with −x, giving y = f(−x). Translating 2 units down then subtracts 2 from the whole expression, producing y = f(−x) − 2.

Question 38Algebra, functions and their graphs

If f(x) = x/(x + 2), what is the value of f(f(−1))?

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

Explanation

First f(−1) = (−1)/(1) = −1. Then f(−1) is applied again, giving the same result −1, so −1 is a fixed point of this function.

Question 39Algebra, functions and their graphs

The functions f(x) = 2x + 1 and g(x) = x² − 3 are defined for all real x. What is the rule for f(g(x))?

  1. 4x² + 4x − 2
  2. 2x² − 5 Correct answer.
  3. 2x² − 6
  4. 4x² − 2

Explanation

The composite f(g(x)) means substituting g(x) into f. This gives 2(x² − 3) + 1 = 2x² − 6 + 1 = 2x² − 5.

Question 40Algebra, functions and their graphs

The graph of y = log₃(x) is transformed to y = log₃(2x). Which single transformation describes this change?

  1. A horizontal dilation by factor 2 about the y-axis
  2. A vertical translation 2 units up
  3. A vertical dilation by factor 2
  4. A horizontal dilation by factor 1/2 about the y-axis Correct answer.

Explanation

Replacing x with 2x in a function compresses the graph horizontally by the factor 1/2, since each output is reached at half the input. Using log laws, log₃(2x) = log₃(x) + log₃(2), so this is also a vertical shift.

Question 41Algebra, functions and their graphs

Solve the inequality x² − 5x + 6 ≤ 0.

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

Explanation

Factorising gives (x − 2)(x − 3) ≤ 0, with boundary values 2 and 3. The parabola opens upward, so it lies below the x-axis between the roots, giving 2 ≤ x ≤ 3.

Question 42Algebra, functions and their graphs

The function h(x) = (x − 1)/(x + 3) has an inverse. What is the rule for h⁻¹(x)?

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

Explanation

Set y = (x − 1)/(x + 3) and swap variables to get x = (y − 1)/(y + 3). Rearranging gives x(y + 3) = y − 1, so y(x − 1) = −1 − 3x, giving y = (1 + 3x)/(1 − x).

Question 43Algebra, functions and their graphs

The graph of y = |x + 2| − 3 has its vertex at which point?

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

Explanation

The absolute value expression |x + 2| is zero when x = −2, which minimises the function. At that point y = 0 − 3 = −3, so the vertex is (−2, −3).

Question 44Algebra, functions and their graphs

For the function p(x) = x³ − 3x + 2, how many distinct real roots does p(x) = 0 have?

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

Explanation

Since p(1) = 0, (x − 1) is a factor, and division gives x³ − 3x + 2 = (x − 1)²(x + 2). The roots are x = 1 (a repeated root) and x = −2, so there are two distinct real roots.

Question 45Algebra, functions and their graphs

The graph of y = 2^x is transformed to y = 2^(x − 3) + 1. What are the horizontal and vertical asymptotes of the new graph?

  1. No vertical asymptote; horizontal asymptote y = 3
  2. No vertical asymptote; horizontal asymptote y = 1 Correct answer.
  3. Vertical asymptote x = 3; horizontal asymptote y = 0
  4. Vertical asymptote x = 3; horizontal asymptote y = 1

Explanation

The exponential function is defined for all real x, so there is no vertical asymptote. The translation upward by 1 moves the horizontal asymptote from y = 0 to y = 1.

Question 46Algebra, functions and their graphs

The function f(x) = x² − 4x + 7 is defined on the restricted domain 0 ≤ x ≤ 5. What is the range of f?

  1. −1 ≤ f(x) ≤ 12
  2. 3 ≤ f(x) ≤ 12 Correct answer.
  3. 3 ≤ f(x) ≤ 7
  4. 7 ≤ f(x) ≤ 12

Explanation

Completing the square gives f(x) = (x − 2)² + 3, so the minimum on the interval is 3 at x = 2. The endpoint values are f(0) = 7 and f(5) = 12, so the maximum is 12.

Question 47Algebra, functions and their graphs

The function f(x) = (x + 1)/(x − 2) is defined for x ≠ 2. Which statement about f is correct?

  1. f is not one-to-one and therefore has no inverse
  2. f has a vertical asymptote at x = −1
  3. f is its own inverse and has a horizontal asymptote y = 1 Correct answer.
  4. f is its own inverse and has a horizontal asymptote y = 2

Explanation

Solving y = (x + 1)/(x − 2) for x gives x = (2y + 1)/(y − 1), which rearranges to (x + 1)/(x − 2), so f is self-inverse. As x grows large the ratio tends to 1, giving the horizontal asymptote y = 1.

Question 48Algebra, functions and their graphs

The equation x³ − 3x² + kx − 2 = 0 has a root at x = 1. What is the value of k?

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

Explanation

By the factor theorem, substituting x = 1 must give zero: 1 − 3 + k − 2 = 0, so k − 4 = 0 and k = 4. This uses the fact that a root corresponds to a zero remainder.

Question 49Algebra, functions and their graphs

The graph of y = f(x) is dilated by factor 2 from the y-axis and then translated 3 units to the right. Which equation describes the new graph?

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

Explanation

A dilation by factor 2 from the y-axis replaces x with x/2, giving y = f(x/2). Translating 3 units right then replaces x with x − 3, producing y = f((x − 3)/2).

Question 50Algebra, functions and their graphs

For the function f(x) = x² − 6x + 5, the inverse relation is not a function. What is the largest domain containing x = 3 on which the inverse of f is a function?

  1. x ≥ 3 Correct answer.
  2. all real x
  3. x ≤ 5
  4. x ≥ 0

Explanation

The parabola has its vertex at x = 3, found from x = −b/(2a) = 3. Restricting to x ≥ 3 makes f strictly increasing, hence one-to-one, so its inverse is a function.