Functions and Graphs
Functions and Graphs tests the connection between algebraic rules and their graphs: domain and range, composition, inverses, transformations, and asymptotic behaviour.
Core rules
- Domain restrictions come from denominators (not zero), even roots (radicand ≥0), and logarithms (argument >0).
- (f∘g)(x)=f(g(x)); composition is not generally commutative.
- To find an inverse, swap x and y, solve for y, and state the restricted domain. Graphs of f and f⁻¹ reflect in y=x.
- Transformations: y=f(x−h)+k shifts right h and up k; y=af(x) stretches vertically by |a|; negative a reflects in the x-axis.
- For rational functions, vertical asymptotes occur where the denominator is zero but the numerator is not; horizontal asymptotes depend on degree comparison.
Worked example
For f(x)=(2x+1)/(x−3), x≠3, find f⁻¹. Set y=(2x+1)/(x−3). Swap: x=(2y+1)/(y−3). Multiply: x(y−3)=2y+1, so xy−3x=2y+1. Collect y: y(x−2)=3x+1, hence y=(3x+1)/(x−2). Thus f⁻¹(x)=(3x+1)/(x−2), x≠2.
Traps
Do not assume f⁻¹ means 1/f. State domains; a formula without its domain can lose marks. When composing, work inside out. A graph crossing a horizontal asymptote is possible; asymptotes describe end behaviour, not forbidden points. Check whether a hole and a vertical asymptote both occur at an excluded x-value.