Modulus Function
Modulus Function
If \(\left| x \right|=c\), where \(c\ge 0\), then \(x=c\) or \(x=-c\)
Graphs of Modulus Functions
\( \qquad\qquad\quad y=mx+c\) e.g. \(y=2x-1\)
| \(y=\left| mx+c \right|\) | \(y=\left| mx+c \right|+p\) |
\( \qquad\qquad\qquad\qquad y=a{{x}^{2}}+bx+c\) e.g. \(y=\left( x-2 \right)\left( x+5 \right)\) | \(y=\left| a{{x}^{2}}+bx+c \right|\)
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Solving modulus equations
- Write the equation as \(\left|\mathrm{f}(x)\right|=\mathrm{g}(x)\).
- Solve \(\mathrm{f}(x)=\mathrm{g}(x)\) and \(\mathrm{f}(x)=-\mathrm{g}(x)\).
- Reject any answer that makes \(\mathrm{g}(x)<0\), because a modulus is never negative.
- For \(\left|\mathrm{f}(x)\right|=\left|\mathrm{g}(x)\right|\), solve \(\mathrm{f}(x)=\mathrm{g}(x)\) or \(\mathrm{f}(x)=-\mathrm{g}(x)\); no rejection step is needed.
Sketching \(y=\left|\mathrm{f}(x)\right|\)
Sketch \(y=\mathrm{f}(x)\), then reflect every part below the \(x\)-axis in the \(x\)-axis. The graph never goes below the \(x\)-axis and has a sharp corner where \(\mathrm{f}(x)=0\).
\(y=\left|2x-1\right|\)
\(y=\left|(x-2)(x+5)\right|\)
Example
Solve \(\left|x-3\right|=2x\).
\(\begin{aligned} x-3&=2x &\quad\text{or}\quad x-3&=-2x \\ x&=-3 &\quad\text{or}\quad x&=1 \end{aligned}\)
\(x=-3\) gives \(2x=-6<0\): rejected. \(x=1\) gives \(\left|1-3\right|=2=2(1)\).
\(x=1\)
Exam check: when the right-hand side contains \(x\), substitute every answer into the original equation. An answer that makes the right-hand side negative must be rejected.
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