A function maps each input (x) to exactly one output (y). Domain = the inputs you're allowed to use. Range = the outputs that come out. Inverse f⁻¹ undoes f — it sends each output back to its input.
f(a) means: take f, substitute x = a, and evaluate. Example: if f(x) = 3x + 2, then f(5) = 3(5) + 2 = 17.
IB-style question — coffee cooling
A cup of coffee's temperature T (°C) after t minutes is modelled by T(t) = 75e⁻⁰.⁰⁴ᵗ + 18, t ≥ 0. Find the temperature 10 minutes after pouring.
Step by step:
Substitute t = 10:
Evaluate on the GDC and round to 3 s.f.:
After 10 minutes, the coffee is about 68.3 °C.
Domain uses the x-values (inputs). Range uses the y-values (outputs). From a graph: read domain along the x-axis, range along the y-axis. Do not swap them.
IB-style question — evaluate f⁻¹(a)
A taxi charges T(d) = 2.40d + 4 dollars for a trip of d km, d ≥ 0. Find T⁻¹(40) and interpret in context.
Step by step:
T⁻¹(40) asks: "for what d does T(d) = 40?" Set T(d) = 40 and solve for d.
Interpret. d is distance in km, so a fare of $40 corresponds to a 15 km trip.
T⁻¹(40) = 15. A trip costing $40 is 15 km long.
IB-style question — find the inverse formula
The formula F = 1.8C + 32 converts temperature from Celsius to Fahrenheit. Find a formula for C in terms of F.
Step by step:
Solve for C. Subtract 32:
Divide both sides by 1.8:
C = (F − 32) / 1.8.
f⁻¹(x) is NOT 1/f(x). The inverse undoes f — it's a different function entirely. The reciprocal is just 1 ÷ f(x). Always write your answer as f⁻¹(x) = … The notation itself earns a mark.
Evaluate f⁻¹(a)? Set f(x) = a, solve for x. Don't derive the formula first. Find the formula? Solve for the other variable. With named variables (F, C, T…) no relabelling needed. Ugly algebra (eˣ, ln, awkward fraction)? Use GDC: Y₁ = f(X), Y₂ = a, intersect. Interpret in context? One sentence in real-world terms. Worth 1 mark on its own. Domain of f⁻¹ = range of f. No formula needed.