Functions: Graphs, Transformations and Inverse Relations
Core concepts
A function is a rule assigning each input in its domain exactly one output. For Ext 1 you must move fluently between algebraic, graphical and inequality descriptions. Master: domain and range restricted by square roots and denominators; odd/even symmetry; absolute value as a piecewise rule; composite functions with correct restricted domains; and inverse relations, remembering that the inverse is a function only when the original is one-to-one.
Traps
- For $y=\sqrt{f(x)}$, require $f(x)\ge 0$; for $\frac{1}{f(x)}$, exclude zeros; combine restrictions for nested expressions.
- Domain of $f(g(x))$ is the set of $x$ in the domain of $g$ with $g(x)$ in the domain of $f$ - not simply the domain of $f$.
- $f^{-1}(x)$ means the inverse function, not $\frac{1}{f(x)}$.
- When sketching $y=|f(x)|$, reflect the part below the axis upward; do not shift.
- Transformations: $y=f(ax+b)$ compresses by factor $a$ before the horizontal shift; order matters.
Rule of thumb
To find an inverse: swap $x$ and $y$, solve for $y$, then state the domain of the inverse as the range of the original. Check with a point: if $(2,5)$ is on $f$, then $(5,2)$ is on $f^{-1}$.
Worked example
For $f(x)=\frac{1}{\sqrt{x-3}}$, require $x-3>0$, so domain $x>3$. As $x\to 3^+$, $f\to\infty$; as $x\to\infty$, $f\to 0^+$. Hence range $y>0$. The graph is the standard $1/\sqrt{x}$ curve shifted right 3, with vertical asymptote $x=3$ and horizontal asymptote $y=0$.