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functions

Notation & inverses, lines, transformations, quadratics, rational functions & asymptotes, graph features.

notation & inverses · lines · transformations · quadratics · exponentials & logs · rational functions · polynomial & modulus graphs · key graph features

Notation & inverses

Notation. \(f:A\to B\); domain \(D_f\), range \(R_f=\{f(x):x\in D_f\}\). Composite \((f\circ g)(x)=f(g(x))\), defined for \(x\in D_g\) with \(g(x)\in D_f\).

Inverse. Exists iff \(f\) is one-to-one; \(f^{-1}(f(x))=x\), \(f(f^{-1}(x))=x\); domain and range swap; graph is the reflection in \(y=x\). Self-inverse: \(f=f^{-1}\).

Symmetry & period. Even: \(f(-x)=f(x)\). Odd: \(f(-x)=-f(x)\). Periodic: \(f(x+T)=f(x)\).

Lines

Forms. \(m=\dfrac{y_2-y_1}{x_2-x_1}\); \(y-y_1=m(x-x_1)\), \(y=mx+c\), \(ax+by+d=0\) with intercepts \(x=-d/a\), \(y=-d/b\).

Parallel & perpendicular. Parallel: \(m_1=m_2\); perpendicular: \(m_1m_2=-1\).

Point–line distance. From \((x_0,y_0)\) to \(ax+by+c=0\): \(d=\dfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}\).

Transformations

Shifts & reflections. \(y=f(x)+b\): up \(b\). \(y=f(x-a)\): right \(a\). \(y=-f(x)\): reflect in the \(x\)-axis. \(y=f(-x)\): reflect in the \(y\)-axis.

Scalings. \(y=pf(x)\): vertical scale \(|p|\). \(y=f(qx)\): horizontal scale \(1/|q|\). \(f(ax+b)=f\bigl(a(x+b/a)\bigr)\): horizontal scale \(1/|a|\), shift \(-b/a\).

Derived graphs. \(y=|f(x)|\): negative \(y\)-values reflected up. \(y=f(|x|)\): right half mirrored left. \(y=1/f(x)\): zeros \(\leftrightarrow\) vertical asymptotes. \(y=[f(x)]^2\ge0\).

Quadratics

Forms & vertex. \(f(x)=ax^2+bx+c=a(x-p)(x-q)=a(x-h)^2+k\). Vertex \((h,k)=\left(-\dfrac b{2a},\ f(-b/2a)\right)\); axis \(x=h\); \(y\)-intercept \(c\); roots by factoring or the quadratic formula; \(a\gt0\) opens up, \(a\lt0\) opens down.

Quadratic inequalities. Solve the roots, then read the sign of the parabola.

Exponentials & logs

Graphs. \(a^x\): domain \(\R\), range \((0,\infty)\), asymptote \(y=0\); increasing for \(a\gt1\), decreasing for \(0\lt a\lt1\). \(\log_a x\): domain \((0,\infty)\), range \(\R\), asymptote \(x=0\); inverse of \(a^x\).

Equations. \(\ln(e^x)=x\), \(e^{\ln x}=x\). Exponential equations: rewrite with the same base or take logs. \(Ae^{kx}+B=C\Rightarrow x=\dfrac1k\ln\left(\dfrac{C-B}{A}\right)\) when valid.

Rational functions & asymptotes

Domain, holes, vertical asymptotes. For \(f=P/Q\): domain excludes zeros of \(Q\). A cancelled common factor gives a removable hole; an uncancelled zero of \(Q\) gives a vertical asymptote.

End asymptotes. \(\deg P\lt\deg Q\Rightarrow y=0\). \(\deg P=\deg Q\Rightarrow y=\) ratio of leading coefficients. \(\deg P=\deg Q+1\Rightarrow\) oblique asymptote \(=\) quotient from division. In general \(P/Q=S+R/Q\), asymptote \(y=S(x)\).

Linear over linear. \(\dfrac{ax+b}{cx+d}\): vertical asymptote \(x=-d/c\), horizontal \(y=a/c\). Intercepts: \(P(x)=0\) and \(f(0)\).

Polynomial & modulus graphs

Polynomial graphs. Degree \(n\), leading coefficient \(a_n\). End behaviour: \(n\) even — both ends take the sign of \(a_n\); \(n\) odd — left end \(-\sgn a_n\), right end \(\sgn a_n\). Roots/factors from \(P(a)=0\); turning points at \(P'(x)=0\); repeated roots: even multiplicity touches, odd crosses. Root sums and products by Vieta (see algebra). Intersections: solve \(f(x)=g(x)\); inequality \(g(x)\ge f(x)\Leftrightarrow g-f\ge0\).

Modulus equations. \(|f(x)|=g(x)\Rightarrow g(x)\ge0\) and \(f(x)=\pm g(x)\). \(|f(x)|\lt g(x)\Rightarrow g(x)\gt0\) and \(-g(x)\lt f(x)\lt g(x)\). \(|f(x)|\gt g(x)\Rightarrow f(x)\gt g(x)\) or \(f(x)\lt-g(x)\), with cases on \(g\). \(|f(x)|=|g(x)|\Rightarrow f^2=g^2\Rightarrow f=\pm g\).

Modulus graphs. \(y=|x-a|+b\): V-shaped, vertex \((a,b)\), slopes \(\pm1\).

Key graph features

Local features. Roots: \(f(x)=0\). \(y\)-intercept: \(f(0)\). Stationary points: \(f'(x)=0\); local extrema via the sign of \(f'\) or \(f''\); inflection where concavity changes.

Asymptotic behaviour. Vertical: \(\lim_{x\to a^\pm}f=\pm\infty\). Horizontal: \(y=L\) where \(\lim_{x\to\pm\infty}f=L\). Oblique: \(y=mx+c\) with \(m=\lim_{x\to\infty}f(x)/x\), \(c=\lim_{x\to\infty}\bigl(f(x)-mx\bigr)\).