Level 1 — RecognitionFunctions

Functions

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Chapter: Functions Level: 1 — Recognition (MCQ + Matching + True/False with justification) Time limit: 20 minutes Total marks: 30

Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. For the function $f(x) = 3x - 5$ , the value of $f(4)$ is: (a) 7 (b) 12 (c) 17 (d) $-2$

Q2. The domain of $f(x) = \dfrac{1}{x-2}$ is: (a) all real numbers (b) $x \neq 0$ (c) $x \neq 2$ (d) $x \geq 2$

Q3. Which relation is NOT a function (fails the vertical line test)? (a) $y = x^2$ (b) $y = 2x + 1$ (c) $x = y^2$ (d) $y = |x|$

Q4. The range of $f(x) = x^2$ (with domain all real numbers) is: (a) all reals (b) $y \geq 0$ (c) $y > 0$ (d) $y \leq 0$

Q5. The graph of $g(x) = f(x) + 3$ is obtained from $f(x)$ by: (a) shift 3 right (b) shift 3 left (c) shift 3 up (d) shift 3 down

Q6. If $f(x) = x^2$ and $g(x) = x + 1$ , then $f(g(x))$ equals: (a) $x^2 + 1$ (b) $(x+1)^2$ (c) $x^2 + x$ (d) $x^3 + 1$

Q7. The inverse of $f(x) = x + 7$ is: (a) $\dfrac{1}{x+7}$ (b) $x - 7$ (c) $7 - x$ (d) $7x$

Q8. Which function is even? (a) $f(x) = x^3$ (b) $f(x) = x^2$ (c) $f(x) = x$ (d) $f(x) = 2x + 1$

Q9. The function $f(x) = 5$ (for all $x$ ) is called a: (a) linear function (b) quadratic function (c) constant function (d) radical function

Q10. A one-to-one function (has an inverse) must pass the: (a) vertical line test only (b) horizontal line test (c) neither test (d) both tests fail

Section B — Matching (1 mark each, Q11–Q15)

Match each function in Column X with its type in Column Y.

Column X	Column Y
Q11. $f(x) = \sqrt{x}$	(A) Quadratic
Q12. $f(x) = \dfrac{x+1}{x-3}$	(B) Linear
Q13. $f(x) = x^2 - 4x + 1$	(C) Radical
Q14. $f(x) = 2x - 9$	(D) Rational
Q15. $f(x) = x^4 - x^3 + 2$	(E) Polynomial (degree 4)

Write your answers as pairs, e.g. Q11–(?).

Section C — True/False WITH Justification (2 marks each, Q16–Q20)

State True or False and give a one-line reason.

Q16. The function $f(x) = |x|$ is odd.

Q17. The graph $y = -f(x)$ is the reflection of $y = f(x)$ in the $x$ -axis.

Q18. For $f(x) = 3x$ , the composition $f(f(x)) = 9x$ .

Q19. A function $f(x) = mx + c$ with $m > 0$ is increasing over all real numbers.

Q20. Every function has an inverse function.

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1. (a) 7 — $f(4) = 3(4) - 5 = 12 - 5 = 7$ . [1]

Q2. (c) $x \neq 2$ — Denominator $x-2=0$ at $x=2$ ; division by zero excluded. [1]

Q3. (c) $x = y^2$ — A vertical line (e.g. $x=1$ ) meets it at $y=\pm1$ : two outputs, fails vertical line test. [1]

Q4. (b) $y \geq 0$ — A square is never negative; smallest value 0 at $x=0$ . [1]

Q5. (c) shift 3 up — Adding a constant outside $f$ raises the graph. [1]

Q6. (b) $(x+1)^2$ — $f(g(x)) = f(x+1) = (x+1)^2$ . [1]

Q7. (b) $x - 7$ — Swap and solve: $y=x+7 \Rightarrow x=y-7$ , so $f^{-1}(x)=x-7$ . [1]

Q8. (b) $f(x)=x^2$ — $f(-x)=(-x)^2=x^2=f(x)$ : even. [1]

Q9. (c) constant function — Output is fixed regardless of input. [1]

Q10. (b) horizontal line test — One-to-one means each output once, so no horizontal line hits twice. [1]

Section B — Matching (1 mark each)

Q11 – (C) radical (square root). [1] Q12 – (D) rational (ratio of polynomials). [1] Q13 – (A) quadratic (degree 2). [1] Q14 – (B) linear (degree 1). [1] Q15 – (E) polynomial degree 4. [1]

Section C — True/False with Justification (2 marks each)

Q16. False [1] — $|{-x}| = |x| = f(x)$ , so it is even, not odd (odd needs $f(-x)=-f(x)$ ). [1 reason]

Q17. True [1] — Negating outputs flips each point to the opposite side of the $x$ -axis (reflection in $x$ -axis). [1 reason]

Q18. True [1] — $f(f(x)) = f(3x) = 3(3x) = 9x$ . [1 reason]

Q19. True [1] — Slope $m>0$ means output rises as $x$ increases throughout $\mathbb{R}$ . [1 reason]

Q20. False [1] — Only one-to-one (injective) functions have inverses; e.g. $f(x)=x^2$ has none over $\mathbb{R}$ . [1 reason]

[
  {"claim":"f(4)=7 for f(x)=3x-5","code":"x=symbols('x'); f=3*x-5; result=(f.subs(x,4)==7)"},
  {"claim":"f(g(x))=(x+1)**2 for f=x^2,g=x+1","code":"x=symbols('x'); f=lambda t:t**2; g=lambda t:t+1; result=(expand(f(g(x)))==expand((x+1)**2))"},
  {"claim":"inverse of x+7 is x-7","code":"x=symbols('x'); result=simplify((x+7)-7 - x)==0 and simplify((x-7)+7 - x)==0"},
  {"claim":"f(f(x))=9x for f(x)=3x","code":"x=symbols('x'); f=lambda t:3*t; result=(simplify(f(f(x))-9*x)==0)"},
  {"claim":"|x| is even not odd","code":"x=symbols('x',real=True); result=(Abs(-x)==Abs(x))"}
]