Functions
Functions on NYS mathematics exams tests relationships across tables, graphs, equations, and contexts: domain and range, transformations, composition, inverses, and rates of change. Expect items that ask you to interpret a feature of a graph or build a function from a description.
Core rules and concepts
- A function assigns exactly one output to each input. Domain exclusions come from denominators equal to zero and even roots of negatives.
- For y=a f(b(x-h))+k, a affects vertical stretch/reflection, b horizontal stretch/compression, h horizontal shift, and k vertical shift.
- Composition (f∘g)(x)=f(g(x)); work inside out. Inverses swap inputs and outputs: start with y=f(x), interchange x and y, solve for y.
- Average rate of change on [a,b] is (f(b)-f(a))/(b-a). Linear functions have constant rate; exponential functions have constant percent change.
- Key graph features include intercepts, intervals of increase/decrease, maxima/minima, and end behavior.
Worked example
Find the inverse of f(x)=3x-5. Write y=3x-5, swap x and y: x=3y-5. Solve: x+5=3y, so y=(x+5)/3. The inverse is f^(-1)(x)=(x+5)/3. Check by composing: f(f^(-1)(x))=3((x+5)/3)-5=x. The original line has domain and range all real numbers, and so does its inverse.
Traps that cost marks
- Treating f(x+h) as f(x)+h; the input is replaced, not added to the output.
- Reversing composition order; f(g(x)) is usually not g(f(x)).
- Giving an inverse without restricting the domain of a non-one-to-one function.
- Reading a graph’s interval or intercept from the wrong axis.