Level 2 — RecallFunctions

Functions

⏱ 30 minutes40 marksprintable — key stays hidden on paper

Level: 2 (Recall — definitions, standard textbook problems, short derivations) Time limit: 30 minutes Total marks: 40

Question 1. [3 marks] Define what is meant by a function. State clearly what the domain, codomain, and range of a function are. (1 mark each)

Question 2. [4 marks] For the function $f(x) = 2x^2 - 3x + 1$ , evaluate: (a) $f(0)$ (1) (b) $f(2)$ (1) (c) $f(-1)$ (1) (d) $f(a+1)$ (leave in simplified form) (1)

Question 3. [3 marks] State the vertical line test. Explain briefly why the graph of $x = y^2$ does not represent $y$ as a function of $x$ .

Question 4. [5 marks] Find the domain of each function. (a) $f(x) = \dfrac{x+2}{x-4}$ (2) (b) $g(x) = \sqrt{x-3}$ (2) (c) $h(x) = 5x^2 + 1$ (1)

Question 5. [4 marks] Match each function to its type (constant, linear, quadratic, rational, radical): (a) $f(x) = 7$ (b) $f(x) = 3x - 2$ (c) $f(x) = \dfrac{1}{x+1}$ (d) $f(x) = \sqrt{2x}$

Question 6. [5 marks] Given $f(x) = x + 3$ and $g(x) = x^2$ , find: (a) $(f \circ g)(x) = f(g(x))$ (2) (b) $(g \circ f)(x) = g(f(x))$ (2) (c) State whether composition of functions is commutative here. (1)

Question 7. [4 marks] Find the inverse function $f^{-1}(x)$ for $f(x) = \dfrac{x-5}{2}$ . Verify your answer by computing $f(f^{-1}(x))$ .

Question 8. [5 marks] Classify each function as even, odd, or neither, showing the algebraic test $f(-x)$ . (a) $f(x) = x^2 - 4$ (2) (b) $f(x) = x^3$ (2) (c) $f(x) = x + 1$ (1)

Question 9. [4 marks] The graph of $y = f(x)$ is transformed. Describe in words the effect on the graph of each: (a) $y = f(x) + 3$ (1) (b) $y = f(x - 2)$ (1) (c) $y = -f(x)$ (1) (d) $y = 2f(x)$ (1)

Question 10. [3 marks] State what it means for a function to be increasing on an interval. For $f(x) = x^2$ , state one interval where it is decreasing and one where it is increasing.

Answer keyMark scheme & solutions

Question 1. [3 marks]

A function is a rule/mapping that assigns to each input exactly one output. (1)
Domain: the set of all permitted input values (x-values). (1)
Codomain: the set into which outputs are declared to map; Range: the set of actual output values produced. (1)

Why: The defining property is uniqueness — one input → one output.

Question 2. [4 marks]

(a) $f(0) = 2(0) - 0 + 1 = 1$ (1)
(b) $f(2) = 2(4) - 3(2) + 1 = 8 - 6 + 1 = 3$ (1)
(c) $f(-1) = 2(1) - 3(-1) + 1 = 2 + 3 + 1 = 6$ (1)
(d) $f(a+1) = 2(a+1)^2 - 3(a+1) + 1 = 2(a^2+2a+1) - 3a - 3 + 1 = 2a^2 + 4a + 2 - 3a - 2 = 2a^2 + a$ (1)

Question 3. [3 marks]

Vertical line test: A graph represents a function iff every vertical line meets the graph at most once. (2)
For $x = y^2$ : a vertical line such as $x = 4$ meets the curve at $y = 2$ and $y = -2$ (two outputs), so it fails the test. (1)

Question 4. [5 marks]

(a) Denominator $\neq 0 \Rightarrow x \neq 4$ ; domain $= \{x \in \mathbb{R} : x \neq 4\}$ (2)
(b) Need $x - 3 \geq 0 \Rightarrow x \geq 3$ ; domain $= [3, \infty)$ (2)
(c) Polynomial: domain $= \mathbb{R}$ (all reals) (1)

Question 5. [4 marks]

(a) constant (1)
(b) linear (1)
(c) rational (1)
(d) radical (1)

Question 6. [5 marks]

(a) $(f\circ g)(x) = f(x^2) = x^2 + 3$ (2)
(b) $(g\circ f)(x) = g(x+3) = (x+3)^2 = x^2 + 6x + 9$ (2)
(c) Not commutative: $x^2 + 3 \neq x^2 + 6x + 9$ in general. (1)

Question 7. [4 marks]

Let $y = \dfrac{x-5}{2}$ . Swap: $x = \dfrac{y-5}{2} \Rightarrow 2x = y - 5 \Rightarrow y = 2x + 5$ . So $f^{-1}(x) = 2x + 5$ . (3)
Check: $f(f^{-1}(x)) = \dfrac{(2x+5)-5}{2} = \dfrac{2x}{2} = x$ . ✓ (1)

Question 8. [5 marks]

(a) $f(-x) = (-x)^2 - 4 = x^2 - 4 = f(x)$ → even (2)
(b) $f(-x) = (-x)^3 = -x^3 = -f(x)$ → odd (2)
(c) $f(-x) = -x + 1$ ; not equal to $f(x)$ nor $-f(x)$ → neither (1)

Question 9. [4 marks]

(a) Shift up by 3 units. (1)
(b) Shift right by 2 units. (1)
(c) Reflection in the x-axis. (1)
(d) Vertical stretch by factor 2. (1)

Question 10. [3 marks]

A function is increasing on an interval if for any $x_1 < x_2$ in it, $f(x_1) < f(x_2)$ (outputs rise as inputs rise). (1)
$f(x)=x^2$ is decreasing on $(-\infty, 0)$ (1) and increasing on $(0, \infty)$ . (1)

[
  {"claim":"f(a+1) simplifies to 2a^2 + a for f(x)=2x^2-3x+1","code":"a=symbols('a'); f=lambda x: 2*x**2-3*x+1; result = simplify(f(a+1) - (2*a**2 + a))==0"},
  {"claim":"f(2)=3 for f(x)=2x^2-3x+1","code":"result = (2*4-3*2+1)==3"},
  {"claim":"composition f(g(x))=x^2+3 and g(f(x))=x^2+6x+9","code":"x=symbols('x'); f=lambda t:t+3; g=lambda t:t**2; result = (simplify(f(g(x))-(x**2+3))==0) and (simplify(g(f(x))-(x**2+6*x+9))==0)"},
  {"claim":"inverse of (x-5)/2 is 2x+5 and f(finv(x))=x","code":"x=symbols('x'); f=lambda t:(t-5)/2; finv=lambda t:2*t+5; result = simplify(f(finv(x))-x)==0"},
  {"claim":"x^3 is odd: f(-x)=-f(x)","code":"x=symbols('x'); result = simplify((-x)**3 + x**3)==0"}
]