Functions
What the domain tests
Functions in AI HL examines modelling and interpretation: domain and range, composite and inverse functions, transformations, and families including quadratic, exponential, logarithmic, rational and sinusoidal. Paper 1 tests algebraic justification; Paper 2 tests graphing-window choices and intersections found by GDC.
Core rules
- For y=af(b(x-h))+k, a is vertical stretch, b horizontal stretch (period divided by b), h horizontal shift, k vertical shift.
- Inverse: swap x and y, then solve; the graph is a reflection in y=x, and domain and range swap.
- Composite f(g(x)) requires the range of g to lie within the domain of f.
- Exponential models y=ab^x+c have horizontal asymptote y=c; logarithmic models need positive arguments.
- Sinusoidal y=A sin(B(x-C))+D has amplitude |A|, period 2π/B, principal axis y=D.
Worked example
Find the inverse of f(x)=(2x+1)/(x-3), x≠3. Write y(x-3)=2x+1, so yx-3y=2x+1. Collect x: x(y-2)=3y+1, hence x=(3y+1)/(y-2). Therefore f^(-1)(x)=(3x+1)/(x-2), with domain x≠2. The vertical asymptote x=3 of f becomes the horizontal asymptote y=3 of the inverse; f has horizontal asymptote y=2, which becomes the inverse’s excluded domain value.
Traps
Reading f^(-1)(x) as 1/f(x); forgetting to restrict the inverse domain; applying transformations in the wrong order; using degrees in a sinusoidal period; accepting a GDC window that misses asymptotes or intersections; ignoring context restrictions such as x≥0.