Topic 2 explores different families of functions, their properties, and transformations. SL covers linear, quadratic, exponential, and logarithmic functions. HL extends to rational functions, factor/remainder theorems, and formal function properties.
A function maps each input to exactly one output. Domain: set of valid inputs; range: set of outputs. Vertical line test for functions. Composite function (f∘g)(x) = f(g(x)) — apply g first. Inverse function f⁻¹: reflects over y = x; (f∘f⁻¹)(x) = x. A function has an inverse only if it is one-to-one (injective).
y = f(x) + a (vertical shift up a), y = f(x − a) (horizontal shift right a), y = af(x) (vertical stretch factor a), y = f(ax) (horizontal compression factor a), y = −f(x) (reflect in x-axis), y = f(−x) (reflect in y-axis). Order matters: apply horizontal changes first inside the function, then vertical changes outside.
f(x) = ax² + bx + c. Vertex form: f(x) = a(x − h)² + k where vertex (h, k). Discriminant Δ = b² − 4ac: Δ > 0 → two real roots, Δ = 0 → one repeated root, Δ < 0 → no real roots. Axis of symmetry: x = −b/(2a). Applications in optimisation and modelling.
f(x) = aˣ and g(x) = logₐx are inverses. Natural exponential f(x) = eˣ and natural logarithm g(x) = ln x. Asymptotic behaviour: exponential has horizontal asymptote y = 0; logarithmic has vertical asymptote x = 0. Modelling growth and decay: N(t) = N₀eᵏᵗ.
Step 1: Write y = f(x). Step 2: Swap x and y to get x = f(y). Step 3: Solve for y — this gives f⁻¹(x). Step 4: Check the domain of f⁻¹ equals the range of f. For example: f(x) = 2x + 3 → y = 2x + 3 → x = 2y + 3 → y = (x−3)/2 → f⁻¹(x) = (x−3)/2.
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