Functions, Graphs and Transformations
The Functions, Graphs and Transformations domain tests whether you can describe a function as a rule, a graph and a mapping, and move fluently between these views. Expect domain and range restrictions, inverse and composite functions, modulus and reciprocal graphs, and transformations of standard curves.
Core rules
- For f: X→Y, domain is allowed inputs, codomain is declared outputs, range is actual outputs. Inverse f^{-1} exists only if f is one-one on its stated domain.
- f^{-1} swaps domain and range; its graph is reflection in y=x. Composite fg means apply g first.
- Transformations: y=af(b(x-c))+d. Inside changes act horizontally and are reversed: b stretches by 1/|b|, c shifts right by c.
- Modulus reflects negative y-values upward; reciprocal has vertical asymptotes at zeros of original and horizontal asymptote at zero if original tends to infinity.
Worked example
Let f(x)=2/(x-1)+3, x>1. Find f^{-1}. Set y=2/(x-1)+3. Then y-3=2/(x-1), so x-1=2/(y-3), x=1+2/(y-3). Since x>1, denominator positive, so y>3. Thus f^{-1}(x)=1+2/(x-3), x>3.
Traps
Forgetting to restrict inverse domain; applying horizontal transformations in wrong order; assuming fg=gf; confusing range with codomain; losing asymptote or endpoint when sketching.