Functions and Their Graphs
This domain tests function concepts: domain and range, function notation, composition, inverses, and transformations, plus the behavior of families-linear, quadratic, polynomial, rational, exponential, and logarithmic. You must connect algebraic form to graphical features: intercepts, vertex, asymptotes, end behavior, and symmetry.
Traps to watch
- Domain of composites and inverses: the domain of $f(g(x))$ excludes values where $g$ is undefined and where $g(x)$ falls outside $f$'s domain. An inverse exists only on a restricted domain if the original is not one-to-one.
- Transformations order: in $y = a,f(b(x-h))+k$, horizontal shifts go opposite the sign, and horizontal stretches/compressions are by $1/b$, not $b$.
- Asymptotes: for rational functions, a horizontal asymptote depends on degree comparison (numerator vs. denominator); a hole appears where a factor cancels.
- Exponential vs. polynomial growth: exponentials eventually outpace any polynomial; don't assume a parabola describes rapid growth.
Worked example / rule of thumb
For $f(x)=\dfrac{x+2}{x^2-4}$, factor the denominator: $(x-2)(x+2)$. The factor $x+2$ cancels, leaving $\dfrac{1}{x-2}$ with a hole at $x=-2$ and a vertical asymptote at $x=2$. The horizontal asymptote is $y=0$ since the denominator's degree exceeds the numerator's. Rule of thumb: factor first, cancel to find holes, then read asymptotes from what remains. For inverses, swap $x$ and $y$, solve, and restrict the domain so the original is one-to-one.