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Sign of a Function Theory
![The sign (positive on negative) of a function is a useful method in solving inequalities.\\ \textbf{Example}\\ By considering the sign of the function \(f(x)=x^3-2 x^2-3 x\) solve the inequality \(x^3-2 x^2-3 x>0\)\\ \textbf{Solution}\\ $\begin{aligned} f(x) & =x^3-2 x^2-3 x \\ & =x\left(x^2-2 x-3\right) \\ & =x(x-3)(x+1) \end{aligned}$\\ To locate critical points let \(f(x)=0\)\\ $\begin{gathered} x(x-3)(x+1)=0 \\ \therefore x=0, x=-1, x=3 . \end{gathered}$\\ Consider the sign of the function on the number line.\\ \begin{center} \resizebox{0.35\textwidth}{!}{ \begin{tikzpicture} \def \xnegative{-4} %negative x value range \def \xpositive{4} %positive x value range \def \xy{\x} %label x or y or any alphabet \draw[latex-latex,line width=0.5mm] (\xnegative.5,0) -- (\xpositive.5,0) node[right] {\Large\(x\)}; \foreach \x in {\xnegative,...,\xpositive} \draw[thin] (\x,4pt )--(\x, -4pt ) node[below=3pt]{\Large\(\xy\)}; \draw[line width=2pt,fill=white] (3,0) circle (0.2); \draw[line width=2pt,fill=white] (-1,0) circle (0.2); \draw[line width=2pt,fill=white] (0,0) circle (0.2); \node[red] at (-0.5,0.5) {\LARGE\(+\)}; \node[red] at (-1.5,0.5) {\LARGE\(-\)}; \node[red] at (1.5,0.5) {\LARGE\(-\)}; \node[red] at (3.5,0.5) {\LARGE\(+\)}; \end{tikzpicture} } \end{center} \(-1<x<0 \cup x>3\)\\ OR \((-1,0) \cup(3,+\infty)\)\\](/media/iqqhxjzc/4770.png)
NSW Y12 Maths - Advanced Graphs and Equations Sign of a Function
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