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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 1:1 if \(f(a) \ne f(b) \> \forall \>\{a,b\} \in \operatorname{dom}(f)\) \\
        \(\implies\) unique \(y\) for each \(x\)
  \item e.g. \(\sin x\) 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 \(\det(\text{first matrix}) = ps-qr\)
      \item Let \(\det = 0\) for \(\{0,\infty\}\) solutions
        or \(\det \ne 0\) for 1 solution
      \item Solve to find variable \\ \\
        \textbf{For infinite/no solutions:}
      \item Substitute variable into both original equations
      \item Rearrange so that LHS of each is the same
      \item \(\begin{aligned}[t]
          \infty \text{ solns: } & \text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x \\
          0 \text{ solns: } & \text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x
      \end{aligned}\)
  \end{enumerate}

  \begin{cas}
    Action \(\rightarrow\) Matrix \(\rightarrow\) Calculation \(\rightarrow\) \texttt{det}
  \end{cas}

  \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
      \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)

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      \begin{figure*}[ht]
        \centering

        \begin{tabularx}{\textwidth}{|r|Y|Y|}

          \hline
          \rowcolor{lblue}
          & \(n\) is even & \(n\) is odd \\ \hline

          \centering \(x^n, n \in \mathbb{Z}^+\) & 

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          \centering \(x^n, n \in \mathbb{Z}^-\) &

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              \addplot[blankplot, domain=0.1:3] {(x^(-1))};
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          \end{tikzpicture}} \\ \hline

          \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &

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            \end{axis}
          \end{tikzpicture}} \\ \hline

        \end{tabularx}
      \end{figure*}

      \section{Polynomials}

      \subsection*{Factor theorem}

      \begin{theorembox}{title=General form \(\beta x + \alpha\)}
        If \(\beta x + \alpha\) is a factor of \(P(x)\), \\
        \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
      \end{theorembox}

      \begin{theorembox}{title=Simple form \(x-a\)}
        If \((x-a)\) is a factor of \(P(x)\), remainder \(R=0\). \\
        \-\hspace{1em}\(\implies P(a)=0\)
      \end{theorembox}

      \subsection*{Remainder theorem}

      \begin{theorembox}{}
        When \(P(x)\) is divided by \(\beta x + \alpha\), the remainder is \(-\dfrac{\alpha}{\beta}\).
      \end{theorembox}

      \subsection*{Rational root theorem}
      Let \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) be a polynomial of degree \(n\) with \(a_i \in \mathbb{Z} \forall a\). Let \(\alpha, \beta \in \mathbb{Z}\) such that their highest common factor is 1 (i.e. relatively prime).

      If \(\beta x + \alpha\) is a factor of \(P(x)\), then \(\beta\) divides \(a_n\) and \(\alpha\) divides \(a_0\) .

      \subsubsection*{Discriminant}
      \[\begin{cases}
        b^2-4ac > 0 & \text{two solutions} \\
        b^2-4ac = 0 & \text{one solution} \\
        b^2-4ac < 0 & \text{no solutions}
      \end{cases}\]
      \begin{warning}
        Flip inequality sign when multiplying by -1
      \end{warning}

      \subsection*{Long division}

      \[ \polylongdiv{x^2+2x+4}{x-1} \]

      \begin{cas}
        Action \(\rightarrow\) Transformation \(\rightarrow\) \texttt{propFrac}
      \end{cas}

      \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}

      \textbf{Linear factorisation}
      \[ x^2 + bx + c = (x+m)(x+n) \]
      \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}

      \subfile{statistics-ref}

    \end{multicols}

\end{document}