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Functions
A-Level H2 Math
Inverse Functions
Need to pass the
horizontal line test
for $f^{-1}$ to exist
$$D_{f^{-1}} = R_f \qquad R_{f^{-1}} = D_f$$
Graphs of inverse functions are reflections about $y = x$
$$f^{-1}f(x) = x \qquad ff^{-1}(x) = x$$
$$D_{f^{-1}f} = D_f \qquad D_{ff^{-1}} = D_{f^{-1}}$$
Solving Equations
Solve $f(x) = f^{-1}(x) \implies$ Solve $f(x) = x$
Solve $ff^{-1}(x) = f^{-1}f(x) \implies D_{f^{-1}} \cap D_f$
Composite Functions
For composite function $gf$ to exist: $R_f \subseteq D_g$
$$gf(x) = g(f(x))$$
$D_{gf} = D_f$ $R_{gf} \subseteq R_g$
Self-Inverse Functions
$f^{-1}(x) = f(x)$ — the function is its own inverse
$$f^2(x) = f^4(x) = f^{\text{even}}(x) = x$$
$$f^3(x) = f^5(x) = f^{\text{odd}}(x) = f(x)$$
Odd & Even Functions
Odd Function
Symmetrical about the
origin
$f(-x) = -f(x)$
Even Function
Symmetrical about the
$y$-axis
$f(-x) = f(x)$
Periodic Functions
$$f(x+a) = f(x)$$
Graph repeats every $a$ units — $a$ is the
period
Formulas for the Singapore A-Level H2 Mathematics syllabus