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Mathematics Methods Level 4 study materials
Mathematics Methods Level 4 study resources

Mathematics Methods Level 4 practice questions

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Question 1Functions, Graphs, and Transformations

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

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

Explanation

Substitute x = 4 into the rule: 3(4) - 7 = 12 - 7 = 5. Function notation means the input value replaces every x in the expression.

Question 2Functions, Graphs, and Transformations

What is the domain of the function g(x) = 1/(x - 5)?

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

Explanation

A fraction is undefined exactly when its denominator equals zero. Setting x - 5 = 0 gives x = 5, so that single value must be excluded while every other real number is allowed.

Question 3Functions, Graphs, and Transformations

The graph of y = f(x) is translated 2 units to the right. Which equation describes the new graph?

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

Explanation

A horizontal shift right by h units replaces x with x - h, so shifting right by 2 gives y = f(x - 2). Adding or subtracting outside the function moves the graph vertically instead.

Question 4Functions, Graphs, and Transformations

Which point is the vertex of the parabola y = (x - 3)^2 + 4?

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

Explanation

Vertex form y = (x - h)^2 + k has vertex (h, k). Here h = 3 and k = 4, so the vertex is (3, 4).

Question 5Functions, Graphs, and Transformations

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

  1. 49
  2. 37 Correct answer.
  3. 20
  4. 19

Explanation

Work from the inside out: g(3) = 6, then f(6) = 36 + 1 = 37. Composite notation f(g(3)) means apply g first and feed the result into f.

Question 6Functions, Graphs, and Transformations

The function f(x) = x^3 is classified as which type of function?

  1. Odd Correct answer.
  2. Neither odd nor even
  3. Even
  4. Constant

Explanation

A function is odd when f(-x) = -f(x) for all x. Since (-x)^3 = -x^3, the cube function satisfies this identity, so it is odd and has rotational symmetry about the origin.

Question 7Functions, Graphs, and Transformations

What is the range of the function y = x^2 - 1 for real x?

  1. y >= -1 Correct answer.
  2. y >= 0
  3. y <= -1
  4. All real y

Explanation

Since x^2 is never negative, its smallest value is 0, giving a minimum output of -1. The parabola opens upward, so the range is all y greater than or equal to -1.

Question 8Functions, Graphs, and Transformations

The graph of y = f(x) has a y-intercept at (0, 5). What is the y-intercept of y = f(x) + 3?

  1. (0, 2)
  2. (3, 5)
  3. (0, 15)
  4. (0, 8) Correct answer.

Explanation

Adding 3 outside the function shifts every point up by 3 units. The point (0, 5) therefore moves to (0, 8), and the x-coordinate stays 0.

Question 9Functions, Graphs, and Transformations

For f(x) = 2x + 1, which expression gives the inverse function f^{-1}(x)?

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

Explanation

Swap x and y in y = 2x + 1 to get x = 2y + 1, then solve for y: y = (x - 1)/2. The inverse undoes the operations in reverse order.

Question 10Functions, Graphs, and Transformations

The graph of y = f(x) is reflected in the x-axis. Which equation gives the reflected graph?

  1. y = -f(x) Correct answer.
  2. y = f(x) - 1
  3. y = |f(x)|
  4. y = f(-x)

Explanation

Reflecting in the x-axis sends each point (x, y) to (x, -y), so the new equation is y = -f(x). Negating the input instead reflects in the y-axis.

Question 11Functions, Graphs, and Transformations

The function h(x) = sqrt(x - 4) is defined for which values of x?

  1. x > 4
  2. x >= 4 Correct answer.
  3. x <= 4
  4. x >= -4

Explanation

A square root requires a non-negative radicand, so x - 4 >= 0, giving x >= 4. The endpoint x = 4 is included because sqrt(0) = 0 is defined.

Question 12Functions, Graphs, and Transformations

The graph of y = f(x) passes through (2, 7). Which point lies on the graph of y = f(x - 3)?

  1. (5, 7) Correct answer.
  2. (2, 10)
  3. (-1, 7)
  4. (2, 4)

Explanation

Replacing x with x - 3 shifts the graph 3 units right, so each point moves to (x + 3, y). The point (2, 7) becomes (5, 7).

Question 13Functions, Graphs, and Transformations

Which statement correctly describes the end behaviour of f(x) = -2x^3 + x?

  1. As x tends to infinity, f(x) tends to positive infinity in both directions.
  2. As x tends to infinity, f(x) tends to negative infinity; as x tends to negative infinity, f(x) tends to positive infinity. Correct answer.
  3. As x tends to infinity, f(x) tends to positive infinity; as x tends to negative infinity, f(x) tends to negative infinity.
  4. As x tends to infinity, f(x) tends to zero in both directions.

Explanation

The leading term -2x^3 controls end behaviour. Its negative coefficient with an odd power means the graph falls to the right and rises to the left, giving the stated limits.

Question 14Functions, Graphs, and Transformations

What is the equation of the horizontal asymptote of y = (3x + 1)/(x - 2)?

  1. y = 2
  2. y = -1/2
  3. y = 3 Correct answer.
  4. y = 0

Explanation

When numerator and denominator have the same degree, the horizontal asymptote is the ratio of leading coefficients. Both are degree 1 with coefficients 3 and 1, giving y = 3.

Question 15Functions, Graphs, and Transformations

If f(x) = x^2 and g(x) = x + 2, what is (f o g)(x)?

  1. x^2 + 4
  2. x^2 + 4x + 4 Correct answer.
  3. x^2 + 2
  4. x^2 + 2x + 2

Explanation

The composition (f o g)(x) means f(g(x)), so substitute g(x) into f: (x + 2)^2 = x^2 + 4x + 4. Expanding the square produces the middle term 4x.

Question 16Functions, Graphs, and Transformations

The graph of y = f(x) has a local maximum at (1, 6). Where is the local maximum of y = f(2x)?

  1. (0.5, 6) Correct answer.
  2. (1, 12)
  3. (2, 6)
  4. (1, 3)

Explanation

Replacing x with 2x compresses the graph horizontally by a factor of 2, so x-coordinates are halved while y-values stay the same. The point (1, 6) becomes (0.5, 6).

Question 17Functions, Graphs, and Transformations

Which function has an inverse that is also a function?

  1. f(x) = x^2 for all real x
  2. f(x) = x^2 for x >= 0 Correct answer.
  3. f(x) = x^2 - 4 for all real x
  4. f(x) = |x| for all real x

Explanation

A function has an inverse function exactly when it is one-to-one, which the horizontal line test confirms. Restricting x^2 to x >= 0 makes it strictly increasing and therefore one-to-one.

Question 18Functions, Graphs, and Transformations

The graph of y = f(x) is transformed to y = 2f(x - 1) + 3. Which sequence of transformations is correct?

  1. Shift right 1, stretch vertically by 2, shift down 3
  2. Shift right 1, stretch horizontally by 2, shift up 3
  3. Shift left 1, stretch vertically by 2, shift up 3
  4. Shift right 1, stretch vertically by 2, shift up 3 Correct answer.

Explanation

Inside the function, x - 1 shifts the graph right 1. The factor 2 outside stretches vertically by 2, and the +3 outside shifts the graph up 3.

Question 19Functions, Graphs, and Transformations

The function f(x) = x^2 + 6x + 5 is written in vertex form. What is its minimum value?

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

Explanation

Complete the square: x^2 + 6x + 5 = (x + 3)^2 - 4. The squared term is at least 0, so the minimum output is -4, occurring at x = -3.

Question 20Functions, Graphs, and Transformations

If f(x) = (x + 1)/(x - 3), what is f^{-1}(x)?

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

Explanation

Set y = (x + 1)/(x - 3) and swap variables: x = (y + 1)/(y - 3). Cross-multiplying gives xy - 3x = y + 1, so y(x - 1) = 3x + 1 and y = (3x + 1)/(x - 1).

Question 21Functions, Graphs, and Transformations

The graph of y = f(x) has a vertical asymptote at x = 2 and a horizontal asymptote at y = 1. Which transformation produces a graph with a vertical asymptote at x = 5?

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

Explanation

A vertical asymptote at x = 2 moves to x = 5 when the graph shifts 3 units right. That requires replacing x with x - 3, giving y = f(x - 3).

Question 22Functions, Graphs, and Transformations

For f(x) = x^2 - 4x + 7, which statement about its inverse relation is true?

  1. The inverse is f^{-1}(x) = sqrt(x - 3) + 2 for all x.
  2. The inverse is not a function because f is not one-to-one on all reals. Correct answer.
  3. The inverse has the same domain as f.
  4. The inverse is a function because f is one-to-one on all reals.

Explanation

The parabola has a line of symmetry at x = 2, so horizontal lines cross it twice and f is not one-to-one. Its inverse relation therefore fails the vertical line test and is not a function unless the domain is restricted.

Question 23Functions, Graphs, and Transformations

The graph of y = f(x) is transformed to y = f(-2x) + 1. Which description is correct?

  1. Reflect in the y-axis, stretch horizontally by factor 2, shift up 1
  2. Reflect in the x-axis, stretch horizontally by factor 2, shift up 1
  3. Reflect in the y-axis, compress horizontally by factor 1/2, shift down 1
  4. Reflect in the y-axis, compress horizontally by factor 1/2, shift up 1 Correct answer.

Explanation

The negative sign on x reflects the graph in the y-axis, and the factor 2 compresses it horizontally by 1/2. The +1 outside shifts the graph up 1 unit.

Question 24Functions, Graphs, and Transformations

Suppose f is an odd function defined for all real x, and f(2) = 5. What is f(-2) + f(0)?

  1. 5
  2. 10
  3. -5 Correct answer.
  4. 0

Explanation

Odd functions satisfy f(-x) = -f(x), so f(-2) = -5. Also f(0) = 0 for any odd function defined at 0, since f(0) = -f(0). The sum is therefore -5.

Question 25Functions, Graphs, and Transformations

The function f(x) = (x^2 - 9)/(x - 3) has a removable discontinuity. What value should be assigned to f(3) to make the function continuous there?

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

Explanation

Factor the numerator as (x - 3)(x + 3), which cancels with the denominator for x not equal to 3, leaving x + 3. The limit as x approaches 3 is 6, so defining f(3) = 6 removes the hole.

Question 26Functions, Graphs, and Transformations

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

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

Explanation

A horizontal shift is applied inside the function argument, and replacing x with x + 4 moves every point 4 units to the left because the input that produces a given output is now 4 less. Vertical changes outside the function move the graph up or down instead.

Question 27Functions, Graphs, and Transformations

What is the domain of f(x) = 1/(x^2 - 9)?

  1. x ≠ ±3 Correct answer.
  2. all real x
  3. x > 3
  4. x ≠ 3 only

Explanation

A rational function is undefined where its denominator equals zero. Solving x^2 - 9 = 0 gives x = 3 or x = -3, so both values must be excluded from the domain.

Question 28Functions, Graphs, and Transformations

The graph of an even function is symmetric with respect to which feature?

  1. the origin
  2. the line y = x
  3. the x-axis
  4. the y-axis Correct answer.

Explanation

An even function satisfies f(-x) = f(x) for every x in its domain, which means the point (-x, y) lies on the graph whenever (x, y) does. This mirror pairing across the vertical axis gives symmetry about the y-axis.

Question 29Functions, Graphs, and Transformations

For f(x) = x^2 + 2x, what is f(x - 1) in expanded form?

  1. x^2 + 2x - 1
  2. x^2 - 1 Correct answer.
  3. x^2 - 2x + 1
  4. x^2 + 1

Explanation

Substituting x - 1 for every x gives (x - 1)^2 + 2(x - 1). Expanding yields x^2 - 2x + 1 + 2x - 2, and the linear terms cancel to leave x^2 - 1.

Question 30Functions, Graphs, and Transformations

The graph of y = f(x) has a vertical asymptote at x = 5. Where is the vertical asymptote of y = f(x + 2)?

  1. x = 7
  2. x = -2
  3. x = 3 Correct answer.
  4. x = 5

Explanation

Replacing x with x + 2 shifts the whole graph 2 units to the left, so every vertical line of interest moves left by 2. The asymptote at x = 5 therefore moves to x = 5 - 2 = 3.

Question 31Functions, Graphs, and Transformations

The graph of y = f(x) passes through the point (4, -1). Which point lies on the graph of y = -f(x) + 2?

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

Explanation

The transformation -f(x) + 2 reflects the graph in the x-axis and then shifts it up 2 units, so the x-coordinate is unchanged. Applying this to the y-value gives -(-1) + 2 = 3, so the image point is (4, 3).

Question 32Functions, Graphs, and Transformations

The function f(x) = x^2 - 4 is restricted to the domain x ≥ 0. What is the domain of f^{-1}?

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

Explanation

The domain of an inverse function equals the range of the original restricted function. On x ≥ 0 the minimum value of x^2 - 4 is -4 and the function increases without bound, so the range is x ≥ -4.

Question 33Functions, Graphs, and Transformations

The graph of y = f(x) has exactly two x-intercepts, at x = -1 and x = 3. What are the x-intercepts of y = f(2x)?

  1. x = -2 and x = 6
  2. x = 1 and x = -3
  3. x = -1/2 and x = 3/2 Correct answer.
  4. x = -1 and x = 3

Explanation

Replacing x with 2x compresses the graph horizontally by a factor of 1/2, so each intercept moves halfway toward the y-axis. Dividing the original intercepts by 2 gives x = -1/2 and x = 3/2.

Question 34Functions, Graphs, and Transformations

For f(x) = |x - 2| + 3, which statement about the graph is correct?

  1. The vertex is at (-2, -3) and the graph opens upward.
  2. The vertex is at (-2, 3) and the graph opens upward.
  3. The vertex is at (2, 3) and the graph opens downward.
  4. The vertex is at (2, 3) and the graph opens upward. Correct answer.

Explanation

The absolute value expression is smallest when x - 2 = 0, so the minimum output is 3 at x = 2, giving the vertex (2, 3). The positive coefficient of the absolute value term makes the V-shape open upward.

Question 35Functions, Graphs, and Transformations

The function f(x) = x^3 - 3x has a local maximum at x = -1 and a local minimum at x = 1. What are the coordinates of the local minimum of y = f(x - 2) - 5?

  1. (3, 3)
  2. (3, -7) Correct answer.
  3. (1, -7)
  4. (-1, -7)

Explanation

The transformation shifts the graph 2 units right and 5 units down, so the point (1, -2) maps to (1 + 2, -2 - 5) = (3, -7). Horizontal shifts act on the x-coordinate with the opposite sign, while the vertical shift applies directly to y.

Question 36Functions, Graphs, and Transformations

The graph of y = f(x) has a horizontal asymptote y = 4 as x approaches infinity. The function g is defined by g(x) = f(3x) - 1. What is the horizontal asymptote of y = g(x) as x approaches infinity?

  1. y = 4
  2. y = 1
  3. y = 3 Correct answer.
  4. y = 12

Explanation

As x approaches infinity, the input 3x also approaches infinity, so f(3x) approaches the same limit 4; horizontal scaling does not change the end value. Subtracting 1 then moves the asymptote down to y = 4 - 1 = 3.

Question 37Polynomial, Rational, and Algebraic Functions

What is the degree of the polynomial 4x^5 - 2x^3 + 7x - 9?

  1. 4
  2. 5 Correct answer.
  3. 9
  4. 3

Explanation

The degree of a polynomial is the highest exponent on the variable with a nonzero coefficient. In 4x^5 - 2x^3 + 7x - 9 the largest exponent is 5, so the degree is 5.

Question 38Polynomial, Rational, and Algebraic Functions

Expand (x + 3)(x - 5).

  1. x^2 + 2x - 15
  2. x^2 - 8x - 15
  3. x^2 - 15
  4. x^2 - 2x - 15 Correct answer.

Explanation

Multiply each term of the first bracket by each term of the second: x^2 - 5x + 3x - 15. Combining the like terms -5x + 3x gives -2x, so the product is x^2 - 2x - 15.

Question 39Polynomial, Rational, and Algebraic Functions

Factor x^2 - 7x + 12.

  1. (x + 3)(x + 4)
  2. (x - 2)(x - 6)
  3. (x - 1)(x - 12)
  4. (x - 3)(x - 4) Correct answer.

Explanation

We need two numbers whose product is +12 and whose sum is -7. The numbers -3 and -4 satisfy both conditions, giving (x - 3)(x - 4).

Question 40Polynomial, Rational, and Algebraic Functions

Simplify (6x^4 y^3) / (2x^2 y).

  1. 3x^2 y^3
  2. 3x^6 y^4
  3. 3x^2 y^2 Correct answer.
  4. 4x^2 y^2

Explanation

Divide the coefficients: 6/2 = 3. For each variable subtract the exponent in the denominator from the exponent in the numerator: x^(4-2) = x^2 and y^(3-1) = y^2, giving 3x^2 y^2.

Question 41Polynomial, Rational, and Algebraic Functions

The rational function f(x) = (2x + 1)/(x - 4) has a vertical asymptote at which value of x?

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

Explanation

A vertical asymptote occurs where the denominator is zero and the numerator is not also zero. Setting x - 4 = 0 gives x = 4, and the numerator there is 9, which is nonzero.

Question 42Polynomial, Rational, and Algebraic Functions

Solve the equation x^2 = 49 for real x.

  1. x = 7 or x = -7 Correct answer.
  2. x = 49 or x = -49
  3. x = 7 only
  4. x = 24.5

Explanation

Taking the square root of both sides of x^2 = 49 gives x = ±7, because both 7^2 and (-7)^2 equal 49. Both signs must be reported for a complete real solution.

Question 43Polynomial, Rational, and Algebraic Functions

Which expression is a polynomial in x?

  1. x^(-2) + 1
  2. sqrt(x) + 5
  3. 2/x + x^2
  4. 3x^3 - x + 2 Correct answer.

Explanation

A polynomial has only nonnegative integer powers of the variable with constant coefficients. The expression 3x^3 - x + 2 fits this definition, while the others contain division by x or fractional and negative powers.

Question 44Polynomial, Rational, and Algebraic Functions

What is the remainder when x^2 + 3x + 5 is divided by x - 1?

  1. 5
  2. 3
  3. 9 Correct answer.
  4. 1

Explanation

By the remainder theorem, the remainder on division by x - 1 equals the polynomial evaluated at x = 1. Substituting gives 1 + 3 + 5 = 9.

Question 45Polynomial, Rational, and Algebraic Functions

A parabola has equation y = 2(x - 3)^2 + 4. What are the coordinates of its vertex?

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

Explanation

The vertex form y = a(x - h)^2 + k has vertex (h, k). Comparing with y = 2(x - 3)^2 + 4 gives h = 3 and k = 4, so the vertex is (3, 4).

Question 46Polynomial, Rational, and Algebraic Functions

Solve the inequality x^2 - 4x < 0.

  1. x < 4
  2. -4 < x < 0
  3. x < 0 or x > 4
  4. 0 < x < 4 Correct answer.

Explanation

Factorising gives x(x - 4) < 0. The parabola opens upward with roots at 0 and 4, so it lies below the x-axis strictly between the roots, giving 0 < x < 4.

Question 47Polynomial, Rational, and Algebraic Functions

The polynomial P(x) = x^3 - 6x^2 + 11x - 6 has a factor (x - 1). Which of the following is the complete factorisation over the integers?

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

Explanation

Dividing by (x - 1) gives the quotient x^2 - 5x + 6, which factorises as (x - 2)(x - 3). Multiplying out confirms the constant term -6 and the x^2 coefficient -6.

Question 48Polynomial, Rational, and Algebraic Functions

What is the horizontal asymptote of f(x) = (3x^2 + 1)/(x^2 - 4)?

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

Explanation

When the numerator and denominator have the same degree, the horizontal asymptote is the ratio of the leading coefficients. Both degrees are 2, and 3/1 = 3, so the asymptote is y = 3.

Question 49Polynomial, Rational, and Algebraic Functions

Simplify the rational expression (x^2 - 9)/(x^2 + x - 12), stating any restriction.

  1. (x + 3)/(x + 4), x not equal to -4
  2. (x - 3)/(x - 4), x not equal to 4 Correct answer.
  3. (x^2 - 9)/(x - 4), x not equal to 4
  4. (x - 3)/(x + 4), x not equal to -4

Explanation

Factorising gives (x - 3)(x + 3) over (x + 4)(x - 3). Cancelling the common factor (x - 3) leaves (x - 3)/(x - 4) only if the denominator is nonzero, so x cannot equal 4.

Question 50Polynomial, Rational, and Algebraic Functions

The polynomial P(x) = 2x^3 - 5x^2 + ax + 4 leaves a remainder of 10 when divided by (x - 2). What is the value of a?

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

Explanation

By the remainder theorem, P(2) = 10. Substituting gives 16 - 20 + 2a + 4 = 10, so 2a = 10, giving a = 5? Checking carefully: 16 - 20 + 4 = 0, so 2a = 10 and a = 5.