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Secondary 4 Elementary Mathematics Algebra Functions Quiz

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Questions

Secondary 4 Elementary Mathematics Quiz - Algebra Functions

Name: ________________________
Class: ________________________
Date: ________________________
Score: ________________________

Duration: 40 minutes
Total Marks: 40

Instructions

Answer all questions.
Show all working clearly in the space provided.
The number of marks for each question is shown in brackets [ ].
Calculators may be used where appropriate.
Give non-exact answers correct to 3 significant figures unless otherwise stated.

Section A: Function Notation and Evaluation (Questions 1–5)

1. Given that $f(x) = 3x^2 - 4x + 7$ , find the value of $f(2)$ .
[2 marks]

2. Given that $g(x) = \dfrac{2x + 5}{x - 3}$ , where $x \ne 3$ , find the value of $g(-1)$ .
[2 marks]

3. The function $h$ is defined as $h : x \mapsto 5 - 2x$ .
(a) Find $h(0)$ .
(b) Find the value of $x$ such that $h(x) = -7$ .
[3 marks]

4. Given that $f(x) = x^2 - 6x + 10$ , find the value of $f(3) - f(1)$ .
[2 marks]

5. The function $p$ is defined as $p(x) = ax + b$ . Given that $p(2) = 11$ and $p(4) = 19$ , find the values of $a$ and $b$ .
[3 marks]

Section B: Composite Functions (Questions 6–10)

6. Given that $f(x) = 2x + 3$ and $g(x) = x^2 - 1$ , find $fg(2)$ .
[2 marks]

7. Given that $f(x) = \dfrac{1}{x}$ and $g(x) = x + 4$ , find $gf(x)$ in terms of $x$ .
[2 marks]

8. Given that $f(x) = 3x - 5$ and $g(x) = 2x + 1$ , find $fg(x)$ and simplify your answer.
[2 marks]

9. Given that $f(x) = x^2 + 2x$ and $g(x) = x - 3$ , find the value of $x$ for which $fg(x) = 0$ .
[3 marks]

10. The functions $f$ and $g$ are defined as $f : x \mapsto 4x - 1$ and $g : x \mapsto \dfrac{x + 2}{3}$ .
(a) Find $fg(x)$ .
(b) Find $gf(x)$ .
[4 marks]

Section C: Inverse Functions (Questions 11–15)

11. Given that $f(x) = 5x - 3$ , find $f^{-1}(x)$ .
[2 marks]

12. Given that $g(x) = \dfrac{2x + 7}{3}$ , find $g^{-1}(x)$ .
[2 marks]

13. The function $h$ is defined as $h : x \mapsto \dfrac{4}{x - 2}$ , where $x \ne 2$ . Find $h^{-1}(x)$ .
[3 marks]

14. Given that $f(x) = 3x + 4$ , find the value of $f^{-1}(10)$ .
[2 marks]

15. The function $f$ is defined as $f : x \mapsto \dfrac{x - 5}{2x + 1}$ , where $x \ne -\dfrac{1}{2}$ .
(a) Find $f^{-1}(x)$ .
(b) State the value of $x$ for which $f^{-1}(x)$ is undefined.
[4 marks]

Section D: Graphs of Functions and Applications (Questions 16–20)

16. The graph of $y = f(x)$ is shown below. It is a parabola with vertex at $(1, -4)$ and passes through the point $(3, 0)$ .
(a) Write down the equation of the axis of symmetry.
(b) State the minimum value of $f(x)$ .
(c) Find the value of $f(0)$ .
[4 marks]

17. The function $f(x) = (x - 2)^2 + 1$ is defined for all real values of $x$ .
(a) Write down the coordinates of the minimum point of the graph of $y = f(x)$ .
(b) State the range of $f(x)$ .
(c) Sketch the graph of $y = f(x)$ for $0 \le x \le 4$ , clearly labelling the vertex and intercepts.
[5 marks]

18. A ball is thrown vertically upwards. Its height $h$ metres above the ground after $t$ seconds is given by $h(t) = 20t - 5t^2$ , for $0 \le t \le 4$ .
(a) Find the height of the ball when $t = 1$ .
(b) Find the maximum height reached by the ball.
(c) Find the value of $t$ when the ball hits the ground.
[5 marks]

19. The function $f$ is defined as $f(x) = 2x^2 - 8x + 5$ .
(a) Express $f(x)$ in the form $a(x - p)^2 + q$ .
(b) Hence, state the coordinates of the minimum point on the graph of $y = f(x)$ .
(c) Find the values of $x$ for which $f(x) = 5$ .
[5 marks]

20. The cost $C$ (in dollars) of producing $n$ items is given by the function $C(n) = 0.5n^2 - 20n + 500$ .
(a) Find the cost of producing 10 items.
(b) Find the number of items that should be produced to minimise the cost.
(c) State the minimum cost.
[5 marks]

End of Quiz

Answers

Secondary 4 Elementary Mathematics Quiz - Algebra Functions

Answer Key

Section A: Function Notation and Evaluation

1. $f(2) = 3(2)^2 - 4(2) + 7 = 3(4) - 8 + 7 = 12 - 8 + 7 = \boxed{11}$
[2 marks] — 1 mark for correct substitution, 1 mark for correct answer.

2. $g(-1) = \dfrac{2(-1) + 5}{-1 - 3} = \dfrac{-2 + 5}{-4} = \dfrac{3}{-4} = \boxed{-\dfrac{3}{4}}$
[2 marks] — 1 mark for correct substitution, 1 mark for correct simplification.

3.
(a) $h(0) = 5 - 2(0) = 5 - 0 = \boxed{5}$
(b) $h(x) = -7 \Rightarrow 5 - 2x = -7 \Rightarrow -2x = -12 \Rightarrow \boxed{x = 6}$
[3 marks] — 1 mark for (a), 2 marks for (b): 1 for setting up equation, 1 for solving.

4. $f(3) = (3)^2 - 6(3) + 10 = 9 - 18 + 10 = 1$
$f(1) = (1)^2 - 6(1) + 10 = 1 - 6 + 10 = 5$
$f(3) - f(1) = 1 - 5 = \boxed{-4}$
[2 marks] — 1 mark for each correct evaluation, or 2 marks for correct final answer with working.

5. $p(2) = 2a + b = 11$ ... (1)
$p(4) = 4a + b = 19$ ... (2)
Subtract (1) from (2): $2a = 8 \Rightarrow a = 4$
Substitute into (1): $2(4) + b = 11 \Rightarrow 8 + b = 11 \Rightarrow b = 3$
$\boxed{a = 4,\ b = 3}$
[3 marks] — 1 mark for setting up equations, 1 mark for solving for $a$ , 1 mark for solving for $b$ .

Section B: Composite Functions

6. $g(2) = (2)^2 - 1 = 4 - 1 = 3$
$fg(2) = f(3) = 2(3) + 3 = 6 + 3 = \boxed{9}$
[2 marks] — 1 mark for finding $g(2)$ , 1 mark for finding $f(g(2))$ .

7. $gf(x) = g\left(\dfrac{1}{x}\right) = \dfrac{1}{x} + 4 = \boxed{\dfrac{1 + 4x}{x}}$ (or $\dfrac{1}{x} + 4$ accepted)
[2 marks] — 1 mark for correct substitution, 1 mark for correct expression.

8. $fg(x) = f(2x + 1) = 3(2x + 1) - 5 = 6x + 3 - 5 = \boxed{6x - 2}$
[2 marks] — 1 mark for correct substitution, 1 mark for simplification.

9. $fg(x) = f(x - 3) = (x - 3)^2 + 2(x - 3) = x^2 - 6x + 9 + 2x - 6 = x^2 - 4x + 3$
Set $fg(x) = 0$ : $x^2 - 4x + 3 = 0$
$(x - 1)(x - 3) = 0$
$\boxed{x = 1 \text{ or } x = 3}$
[3 marks] — 1 mark for finding $fg(x)$ , 1 mark for setting up equation, 1 mark for solving.

10.
(a) $fg(x) = f\left(\dfrac{x + 2}{3}\right) = 4\left(\dfrac{x + 2}{3}\right) - 1 = \dfrac{4x + 8}{3} - 1 = \dfrac{4x + 8 - 3}{3} = \boxed{\dfrac{4x + 5}{3}}$
(b) $gf(x) = g(4x - 1) = \dfrac{(4x - 1) + 2}{3} = \dfrac{4x + 1}{3}$
[4 marks] — 2 marks for (a), 2 marks for (b): 1 mark for substitution, 1 mark for simplification each.

Section C: Inverse Functions

11. Let $y = 5x - 3$ . Swap $x$ and $y$ : $x = 5y - 3$
Solve for $y$ : $5y = x + 3 \Rightarrow y = \dfrac{x + 3}{5}$
$f^{-1}(x) = \boxed{\dfrac{x + 3}{5}}$
[2 marks] — 1 mark for swapping variables, 1 mark for solving.

12. Let $y = \dfrac{2x + 7}{3}$ . Swap: $x = \dfrac{2y + 7}{3}$
$3x = 2y + 7 \Rightarrow 2y = 3x - 7 \Rightarrow y = \dfrac{3x - 7}{2}$
$g^{-1}(x) = \boxed{\dfrac{3x - 7}{2}}$
[2 marks] — 1 mark for swapping, 1 mark for solving.

13. Let $y = \dfrac{4}{x - 2}$ . Swap: $x = \dfrac{4}{y - 2}$
$x(y - 2) = 4 \Rightarrow xy - 2x = 4 \Rightarrow xy = 4 + 2x \Rightarrow y = \dfrac{4 + 2x}{x}$
$h^{-1}(x) = \boxed{\dfrac{4 + 2x}{x}}$ (or $\dfrac{2x + 4}{x}$ )
[3 marks] — 1 mark for swapping, 1 mark for cross-multiplying, 1 mark for isolating $y$ .

14. $f^{-1}(10)$ : Let $f(x) = 10 \Rightarrow 3x + 4 = 10 \Rightarrow 3x = 6 \Rightarrow x = 2$
Alternatively, find $f^{-1}(x) = \dfrac{x - 4}{3}$ , then $f^{-1}(10) = \dfrac{10 - 4}{3} = \dfrac{6}{3} = \boxed{2}$
[2 marks] — 1 mark for method, 1 mark for answer.

15.
(a) Let $y = \dfrac{x - 5}{2x + 1}$ . Swap: $x = \dfrac{y - 5}{2y + 1}$
$x(2y + 1) = y - 5 \Rightarrow 2xy + x = y - 5 \Rightarrow 2xy - y = -5 - x$
$y(2x - 1) = -5 - x \Rightarrow y = \dfrac{-5 - x}{2x - 1} = \boxed{\dfrac{x + 5}{1 - 2x}}$ (or equivalent)
(b) $f^{-1}(x)$ is undefined when $2x - 1 = 0 \Rightarrow \boxed{x = \dfrac{1}{2}}$
[4 marks] — 2 marks for (a) (swap and solve), 2 marks for (b) (1 for setting denominator to 0, 1 for solving).

Section D: Graphs of Functions and Applications

16.
(a) Axis of symmetry: $\boxed{x = 1}$
(b) Minimum value: $\boxed{-4}$
(c) Since vertex is $(1, -4)$ and passes through $(3, 0)$ :
Using $y = a(x - 1)^2 - 4$ , substitute $(3, 0)$ : $0 = a(2)^2 - 4 \Rightarrow 4a = 4 \Rightarrow a = 1$
$f(x) = (x - 1)^2 - 4$ , so $f(0) = (0 - 1)^2 - 4 = 1 - 4 = \boxed{-3}$
[4 marks] — 1 mark each for (a), (b), (c) setup, (c) answer.

17.
(a) Minimum point: $\boxed{(2, 1)}$
(b) Range: $\boxed{y \ge 1}$ (or $[1, \infty)$ )
(c) Sketch: Parabola opening upwards with vertex at $(2, 1)$ , $y$ -intercept at $(0, 5)$ , $x$ -intercept: none (since minimum is 1).
[5 marks] — 1 mark for (a), 1 mark for (b), 3 marks for sketch (shape, vertex, intercepts).

18.
(a) $h(1) = 20(1) - 5(1)^2 = 20 - 5 = \boxed{15}$ metres
(b) $h(t) = -5t^2 + 20t = -5(t^2 - 4t) = -5[(t - 2)^2 - 4] = -5(t - 2)^2 + 20$
Maximum height = $\boxed{20}$ metres (at $t = 2$ )
(c) Ball hits ground when $h(t) = 0$ : $20t - 5t^2 = 0 \Rightarrow 5t(4 - t) = 0$
$t = 0$ (start) or $\boxed{t = 4}$ seconds
[5 marks] — 1 mark for (a), 2 marks for (b) (completing square or derivative), 2 marks for (c).

19.
(a) $f(x) = 2x^2 - 8x + 5 = 2(x^2 - 4x) + 5 = 2[(x - 2)^2 - 4] + 5 = 2(x - 2)^2 - 8 + 5 = \boxed{2(x - 2)^2 - 3}$
(b) Minimum point: $\boxed{(2, -3)}$
(c) $f(x) = 5 \Rightarrow 2x^2 - 8x + 5 = 5 \Rightarrow 2x^2 - 8x = 0 \Rightarrow 2x(x - 4) = 0$
$\boxed{x = 0 \text{ or } x = 4}$
[5 marks] — 2 marks for (a) (completing square), 1 mark for (b), 2 marks for (c).

20.
(a) $C(10) = 0.5(10)^2 - 20(10) + 500 = 0.5(100) - 200 + 500 = 50 - 200 + 500 = \boxed{350}$ dollars
(b) $C(n) = 0.5n^2 - 20n + 500 = 0.5(n^2 - 40n) + 500 = 0.5[(n - 20)^2 - 400] + 500 = 0.5(n - 20)^2 - 200 + 500 = 0.5(n - 20)^2 + 300$
Minimum when $n = \boxed{20}$ items
(c) Minimum cost = $\boxed{300}$ dollars
[5 marks] — 1 mark for (a), 2 marks for (b) (completing square), 2 marks for (c) (substituting or reading from vertex form).

End of Answer Key