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\fancyhead[LO,LE]{Year 12 Methods}
\fancyhead[CO,CE]{Andrew Lorimer}
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\begin{document}

\title{\vspace{-20mm}Year 12 Methods}
\author{Andrew Lorimer}
\date{}
\maketitle

\begin{multicols}{2}

  \section{Functions}

  \begin{itemize}
      \tightlist
    \item vertical line test
    \item each \(x\) value produces only one \(y\) value
  \end{itemize}

  \subsection*{One to one functions}

  \begin{itemize} \tightlist
    \item
      \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
      \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
      \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
      \(x^3\) is)
    \item
      horizontal line test
    \item
      if not one to one, it is many to one
  \end{itemize}

      \subsection*{Odd and even functions}

      \begin{align*}
        \text{Even:}&& f(x)  &= f(-x) \\
        \text{Odd:} && -f(x) &= f(-x)
      \end{align*}

      Even \(\implies\) symmetrical across \(y\)-axis \\
      \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
      For \(x^n\), parity of \(n \equiv\) parity of function

      \begin{tabularx}{\columnwidth}{XX}
        \textbf{Even:} & \textbf{Odd:} \\
        \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)};  \end{axis}\end{tikzpicture} &
          \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)};  \end{axis}\end{tikzpicture}
      \end{tabularx}

  \subsection*{Inverse functions}

  \begin{itemize} \tightlist
    \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
    \item \(f\) must be one to one
    \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
    \item Represents reflection across \(y=x\)
    \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
    \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
      \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
    \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
  \end{itemize}

  \subsubsection*{Finding \(f^{-1}\)}

  \begin{enumerate} \tightlist
    \item Let \(y=f(x)\)
    \item Swap \(x\) and \(y\) (``take inverse''
    \item Solve for \(y\) \\
      Sqrt: state \(\pm\) solutions then restrict
    \item State rule as \(f^{-1}(x)=\dots\)
    \item For inverse \emph{function}, state in function notation
  \end{enumerate}
      
  \subsection*{Simultaneous equations (linear)}

  \begin{itemize} \tightlist
    \item \textbf{Unique solution} - lines intersect at point
    \item \textbf{Infinitely many solutions} - lines are equal
    \item \textbf{No solution} - lines are parallel
  \end{itemize}

  \subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
    where all coefficients are known except for one, and \(a, b\) are known

    \begin{enumerate} \tightlist
      \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
        \item Find determinant of first matrix: \(\Delta = ps-qr\)
        \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
          or let \(\Delta \ne 0\) for one unique solution.
        \item Solve determinant equation to find variable \\
          \textbf{For infinite/no solutions:}
        \item Substitute variable into both original equations
        \item Rearrange equations so that LHS of each is the same
        \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
          \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
    \end{enumerate}

    \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}

    \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}

      \begin{itemize} \tightlist
        \item Use elimination
        \item Generate two new equations with only two variables
        \item Rearrange \& solve
        \item Substitute one variable into another equation to find another variable
      \end{itemize}

\subsection*{Piecewise functions}

\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]

\textbf{Open circle:} point included\\
\textbf{Closed circle:} point not included

\subsection*{Operations on functions}

For \(f \pm g\) and \(f \times g\):
\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)

Addition of linear piecewise graphs: add \(y\)-values at key points

Product functions:

\begin{itemize}
\tightlist
\item
  product will equal 0 if \(f=0\) or \(g=0\)
\item
  \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
\end{itemize}

\subsection*{Composite functions}

\((f \circ g)(x)\) is defined iff
\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)


      \pgfplotsset{every axis/.append style={ ticks=none, xlabel=, ylabel=, }} % remove axis labels & ticks
      \begin{table*}[ht]
        \centering
        \begin{tabu} to \textwidth {@{} X[0.3,r] *2{|X[c,m]}@{}}
          & \(n\) is even & \(n\) is odd \\ \tabucline{1pt} 
          \(x^n, n \in \mathbb{Z}^+\) & 
          \vspace{1em}\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^2)};  \end{axis}\end{tikzpicture} &
            \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^3)};  \end{axis}\end{tikzpicture} \\
              \(x^n, n \in \mathbb{Z}^-\) &
              \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-4,  xmax=4, ymax=8, ymin=-0, scale=0.4, smooth] \addplot[orange, mark=none, samples=100] {(x^(-2))};  \end{axis}\end{tikzpicture} &
                \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none, domain=-3:-0.1] {(x^(-1))}; \addplot[orange, mark=none, domain=0.1:3] {(x^(-1))};  \end{axis}\end{tikzpicture} \\
                  \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
                  \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-1,  xmax=5, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^(1/2))};  \end{axis}\end{tikzpicture} &
                    \begin{tikzpicture}
                      \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4]
                        \addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
                      \end{axis}
                    \end{tikzpicture}
        \end{tabu}
        \hrule
      \end{table*}
      \pgfplotsset{every axis/.append style={ xlabel=\(x\), ylabel=\(y\) }} % put axis labels back

      \section{Polynomials}

      \subsection*{Linear equations}

      \subsubsection*{Forms}

      \begin{itemize}
          \tightlist
        \item \(y=mx+c\)
        \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
        \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
      \end{itemize}

      \subsubsection*{Line properties}

      Parallel lines: \(m_1 = m_2\)\\
      Perpendicular lines: \(m_1 \times m_2 = -1\)\\
      Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

      \subsection*{Quadratics}
      \setlength{\abovedisplayskip}{1pt}
      \setlength{\belowdisplayskip}{1pt}
      \[ x^2 + bx + c = (x+m)(x+n) \]
      \hfill where \(mn=c, \> m+n=b\)

      \textbf{Difference of squares}
      \[ a^2 - b^2 = (a-b)(a+b) \]
      \textbf{Perfect squares}
      \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
      \textbf{Completing the square}
      \begin{align*}
        x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
        ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
      \end{align*}
      \textbf{Quadratic formula}
      \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
      \hfill (Discriminant \(\Delta=b^2-4ac\))

      \subsection*{Cubics}

      \textbf{Difference of cubes}
      \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
      \textbf{Sum of cubes}
      \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
      \textbf{Perfect cubes}
      \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]

      \[ y=a(bx-h)^3 + c \]

      \begin{itemize}
          \tightlist
        \item
          \(m=0\) at \emph{stationary point of inflection}
          (i.e.~(\({h \over b}, k)\))
        \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
        \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
        \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
      \end{itemize}

      \subsection*{Quartic graphs}

      \subsubsection*{Forms of quartic equations}

      \(y=ax^4\)\\
      \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
      \(y=ax^4+cd^2 (c \ge 0)\)\\
      \(y=ax^2(x-b)(x-c)\)\\
      \(y=a(x-b)^2(x-c)^2\)\\
      \(y=a(x-b)(x-c)^3\)

      \input{transformations}
      \input{stuff}
      \input{circ-functions}
      \input{calculus}

    \end{multicols}
  \end{document}
