T4T1 Review — Functions and Relations

Mixed review covering L40 (Functions & Function Notation), L41 (Domain, Range & Types of Functions) and L42 (Inverse & Composite Functions).

1. Functions and notation. Fluency

   For f(x) = 4x − 3, find:

   • (a) f(5)
   • (b) f(−2)
   • (c) f(a − 1)
   • (d) x if f(x) = 21

2. Function or not? Fluency

   • (a) {(1,3), (2,5), (3,7), (1,9)} — is this a function?
   • (b) y = 7 (for all x) — is this a function?
   • (c) x = −2 (a vertical line) — is this a function?
   • (d) y = x² − 5x + 6 — is this a function?

3. Natural domain. Fluency

   • (a) f(x) = √(2x + 8)
   • (b) g(x) = 5/(x − 2)
   • (c) h(x) = √(16 − x²)
   • (d) k(x) = 1/√(x + 5)

4. Finding inverses. Fluency

   • (a) f(x) = 6x + 2
   • (b) g(x) = (x + 4)/3
   • (c) h(x) = x³ − 2
   • (d) If f(3) = 10 and f(7) = 4, find f⁻¹(10) and f⁻¹(4).

5. Graph reading — function properties. Understanding

   The graph shows f(x) = −(x − 2)² + 4.

   • (a) State the domain and range of f.
   • (b) State the x-intercepts and y-intercept.
   • (c) Solve f(x) ≥ 0. Write the solution as an inequality.
   • (d) Does f have an inverse function over all reals? If not, suggest a domain restriction.

6. Composite functions. Understanding

   Let f(x) = x + 5 and g(x) = x².

   • (a) Find f(g(x)) and g(f(x)).
   • (b) Find f(g(3)) and g(f(3)).
   • (c) Find x such that f(g(x)) = g(f(x)).
   • (d) Find x such that f(g(x)) = 14.

7. Range and inverse. Understanding

   • (a) f(x) = 3x² − 5. State the range, then restrict the domain to x ≥ 0 and find f⁻¹(x).
   • (b) f has domain [−3, 6] and rule f(x) = 2x − 1. Find the range of f, and the domain and range of f⁻¹.
   • (c) Find the inverse of f(x) = (3x − 1)/(x + 2). What x-value is excluded from the domain of f? What y-value is excluded from the range?
   • (d) Verify your answer to (c) by computing f(f⁻¹(x)).

8. Real-world functions. Understanding

   A car's fuel consumption is f(d) = d/12 litres, where d is distance in km. Petrol costs $2.20/litre, so the cost function is c(v) = 2.20v where v is volume in litres.

   • (a) Find c(f(d)) and interpret its meaning.
   • (b) How much does it cost to drive 300 km?
   • (c) Find the inverse of c(f(d)) and interpret it.
   • (d) How far can you drive on a $40 petrol budget?

9. Decomposition and domain. Problem Solving

   • (a) Express h(x) = 1/√(x + 3) as f(g(x)). State f and g, and find the natural domain of h.
   • (b) Let f(x) = √x (domain x ≥ 0) and g(x) = 4 − x². Find the domain of f(g(x)).
   • (c) If g(x) = 3x − 2 and f(g(x)) = 9x² − 12x + 5, find f(x).
   • (d) Show that if f and g are both one-to-one, then f(g(x)) is also one-to-one.

10. Modelling with functions. Problem Solving

    A farmer's crop yield (tonnes) is Y(r) = −0.2r² + 8r − 60 where r is rainfall (mm), valid for 10 ≤ r ≤ 30.

    • (a) State the domain in context. Find Y(10) and Y(30).
    • (b) Find the rainfall that maximises yield and the maximum yield.
    • (c) State the range of Y over the given domain.
    • (d) Find the inverse Y⁻¹ restricted to r ≥ 20. What does it represent, and what is Y⁻¹(12)?

11. Multi-step function problem. Problem Solving

    • (a) f(x) = ax + b with f(2) = 7 and f(5) = 13. Find a and b, then find f⁻¹(x).
    • (b) For f(x)=2x+3 from (a) and g(x)=x−1, find all x where f(g(x))=g(f(x)).
    • (c) h(x) = f(g(f(x))). Simplify h(x) and find h⁻¹(x).
    • (d) Verify: h⁻¹(h(x)) = x.

12. Extended problem. Problem Solving

    A temperature conversion: Celsius to Fahrenheit is F(c) = (9c/5) + 32. Fahrenheit to Kelvin is K(f) = f + 273.15.

    • (a) Find K(F(c)), the composite function from Celsius to Kelvin.
    • (b) Convert 100°C to Kelvin using the composite.
    • (c) Find F⁻¹(f), the inverse of F (converting Fahrenheit back to Celsius).
    • (d) The standard result is K = C + 273.15. Show this matches your composite.