Mathematics AA: Higher Level
Functions — Topic 2
- 1.
Show that (x − 2) is a factor of p(x) = x³ − 3x² − 4x + 12, and hence fully factorise p(x).
[5 marks] · no calculator - 2.
Marking analysis: A learner attempts the following task: “Show that (x − 2) is a factor of p(x) = x³ − 3x² − 4x + 12, and hence fully factorise p(x).” Their response addresses only this point: “Applies the factor theorem by evaluating p(2) = 8 − 12 − 8 + 12 = 0.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] · no calculator - 3.
The rational function f(x) = (2x + 1)/(x − 3) has a vertical and a horizontal asymptote. State the equation of each asymptote.
[3 marks] · no calculator - 4.
Marking analysis: A learner attempts the following task: “The rational function f(x) = (2x + 1)/(x − 3) has a vertical and a horizontal asymptote. State the equation of each asymptote.” Their response addresses only this point: “Identifies the vertical asymptote where the denominator is zero: x = 3.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 5.
The cubic equation 2x³ − 5x² + 3x − 1 = 0 has roots α, β, γ. Find α + β + γ and αβγ.
[4 marks] · no calculator - 6.
Marking analysis: A learner attempts the following task: “The cubic equation 2x³ − 5x² + 3x − 1 = 0 has roots α, β, γ. Find α + β + γ and αβγ.” Their response addresses only this point: “States the relations for ax³ + bx² + cx + d = 0: sum of roots = −b/a, product of roots = −d/a.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 7.
The function f(x) = √(x + 4) has domain x ≥ −4. Find f⁻¹(x) and state its domain and range.
[5 marks] · no calculator - 8.
Marking analysis: A learner attempts the following task: “The function f(x) = √(x + 4) has domain x ≥ −4. Find f⁻¹(x) and state its domain and range.” Their response addresses only this point: “Writes y = √(x + 4) and swaps x and y: x = √(y + 4).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] · no calculator - 9.
Solve the inequality (x − 1)(x + 2)(x − 3) ≥ 0 using a sign diagram.
[5 marks] · no calculator - 10.
Marking analysis: A learner attempts the following task: “Solve the inequality (x − 1)(x + 2)(x − 3) ≥ 0 using a sign diagram.” Their response addresses only this point: “Identifies the critical values x = −2, 1, 3 where the expression equals zero.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] · no calculator - 11.
Show algebraically that f(x) = (x + 1)/(x − 1), x ≠ 1, is a self-inverse function.
[5 marks] · no calculator - 12.
Marking analysis: A learner attempts the following task: “Show algebraically that f(x) = (x + 1)/(x − 1), x ≠ 1, is a self-inverse function.” Their response addresses only this point: “States that f is self-inverse if f(f(x)) = x for all x in the domain.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] · no calculator - 13.
Given f(x) = √x and g(x) = x − 4, find the domain of the composite function (f ∘ g)(x).
[4 marks] · no calculator - 14.
Marking analysis: A learner attempts the following task: “Given f(x) = √x and g(x) = x − 4, find the domain of the composite function (f ∘ g)(x).” Their response addresses only this point: “Forms the composite (f ∘ g)(x) = f(g(x)) = √(x − 4).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 15.
Determine algebraically whether f(x) = x³ − 2x is odd, even, or neither.
[4 marks] · no calculator - 16.
Marking analysis: A learner attempts the following task: “Determine algebraically whether f(x) = x³ − 2x is odd, even, or neither.” Their response addresses only this point: “States the tests: f is even if f(−x) = f(x) for all x; f is odd if f(−x) = −f(x) for all x.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 17.
Express (5x − 3)/((x − 1)(x + 2)) in the form A/(x − 1) + B/(x + 2), finding the values of A and B.
[5 marks] · no calculator - 18.
Marking analysis: A learner attempts the following task: “Express (5x − 3)/((x − 1)(x + 2)) in the form A/(x − 1) + B/(x + 2), finding the values of A and B.” Their response addresses only this point: “States 5x − 3 = A(x + 2) + B(x − 1).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] · no calculator - 19.
The graph of y = f(x) is transformed to give y = 2f(x − 3) − 1. Describe, in order, the three transformations applied to the graph of y = f(x).
[3 marks] · no calculator - 20.
Marking analysis: A learner attempts the following task: “The graph of y = f(x) is transformed to give y = 2f(x − 3) − 1. Describe, in order, the three transformations applied to the graph of y = f(x).” Their response addresses only this point: “States a horizontal translation of 3 units in the positive x-direction (right), giving f(x − 3).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 21.
Solve the equation 3^(2x−1) = 27^(x+2), giving x as an exact value.
[4 marks] · no calculator - 22.
Marking analysis: A learner attempts the following task: “Solve the equation 3^(2x−1) = 27^(x+2), giving x as an exact value.” Their response addresses only this point: “Rewrites 27 as 3³, so 27^(x+2) = 3^(3(x+2)).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 23.
The rational function f(x) = (x² + 1)/(x − 2) has an oblique (slant) asymptote. Use polynomial division to find its equation.
[4 marks] · no calculator - 24.
Marking analysis: A learner attempts the following task: “The rational function f(x) = (x² + 1)/(x − 2) has an oblique (slant) asymptote. Use polynomial division to find its equation.” Their response addresses only this point: “Divides x² + 1 by x − 2 using polynomial long division.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 25.
Solve the equation |2x − 5| = 7.
[3 marks] · no calculator - 26.
Marking analysis: A learner attempts the following task: “Solve the equation |2x − 5| = 7.” Their response addresses only this point: “Sets up the two cases: 2x − 5 = 7 and 2x − 5 = −7.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 27.
The polynomial f(x) = x³ + ax² − 5x + 6 has (x − 2) as a factor. Find the value of a.
[3 marks] - 28.
Marking analysis: A learner attempts the following task: “The polynomial f(x) = x³ + ax² − 5x + 6 has (x − 2) as a factor. Find the value of a.” Their response addresses only this point: “States the factor theorem: since (x − 2) is a factor, f(2) = 0.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 29.
Given f(x) = 2x + 1 and g(x) = x² − 3, find (g∘f)(x) and hence solve (g∘f)(x) = 6.
[4 marks] - 30.
Marking analysis: A learner attempts the following task: “Given f(x) = 2x + 1 and g(x) = x² − 3, find (g∘f)(x) and hence solve (g∘f)(x) = 6.” Their response addresses only this point: “Forms the composite: (g∘f)(x) = g(2x + 1) = (2x + 1)² − 3.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]