The Modulus Function

Modulus Notation

The following notation is relevant to the modulus:

• The modulus of a number, e.g. x is written as |x|.
• Generally, |x| = x for x \geq 0, and |x| = -x for x <0
• Functions also have a modulus: e.g. if f(x) = -2, then |f(x)| = 2
• |f(x)| = f(x) when f(x) \geq 0 and |f(x)| = -f(x) when f(x)<0
• If the modulus is inside the function, e.g. f(|x|), then you apply the modulus to the x-value before applying the function, i.e. f(|-4|) = f(4)

Graphs of Modulus Functions – Straight Lines

There are 3 types of modulus graphs that you may be asked to draw:

1. y = |f(x)| – all negative values of f(x) are made positive, by reflecting the negative section of the graph of f(x) in the x-axis. This restricts the range to |f(x)| \geq 0 (or a subset within |f(x)| \geq 0, e.g. |f(x)| \geq 2.
2. y = f(|x|) – the negative x-values give the same result as the corresponding positive x-values, so the graph of f(x) for x \geq 0 is reflected in the y-axis, for negative x-values.
3. y = |f(-x)| – the x-values swap sign (i.e. from positive to negative of from negative to positive), so the graph of f(x) is reflected in the y-axis. Then, all negative values of f(x) are made positive by reflecting the negative section of the graph of f(x) in the x-axis. The range is restricted, as with y = |f(x)|.

The best and easiest way to draw these graphs is to plot the graph of y = f(x) first, and then reflect it in the appropriate axis or axes.

Example: For f(x) = 2x-1, sketch the graphs of

\begin{aligned} y &= |f(x)| \\ y &= f(|x|) \\ y &= f|(-x)| \end{aligned}

Graphs of Modulus Functions – Quadratics and Cubics etc.

For modulus graphs where the function is a quadratic or cubic etc. the same rules apply as for straight lines – however, sketching them will be a little bit harder.

Example: For f(x) = x^2-2x, sketch the graphs of

\begin{aligned} y &= |f(x)| \\ y &= f(|x|) \\ y &= |f(-x)| \end{aligned}

Solving Modulus Equations Graphically

To solve modulus equations of the form |f(x)| = n or |f(x)| = |g(x)|, you can solve them graphically, using the following method:

Step 1: Sketch the graphs of y = |f(x)| and y = n, on the same pair of axes.

Step 2: Work out the ranges of x for which f(x) \geq 0 and f(x) < 0 from the graph.

e.g. f(x) \geq 0   for x \leq \textcolor{red}{a} or x \geq \textcolor{blue}{b}   and   f(x) < 0  for \textcolor{red}{a} < x < \textcolor{blue}{b}

Step 3: Use step 2 to write 2 new equations, one that holds for each range of x:

f(x) = n   for x \leq \textcolor{red}{a} or x \geq \textcolor{blue}{b}

- f(x) = n   for \textcolor{red}{a} < x < \textcolor{blue}{b}

Step 4: Solve each equation in turn and check that the solutions are valid, and remove any that are outside the range of x for that equation.

Step 5: Check that the solutions look correct, by looking at the graph.

Note: Use the same method for |f(x)| = |g(x)|, by replacing n with g(x).

Solving Modulus Equations Algebraically

For equations of the form |f(x)| = n and |f(x)| = g(x) you can solve them algebraically instead of graphically – if you feel that you understand the topic well enough.

Example: Solve |2x+2| = x+4

Step 1: Solve for positive values:

\begin{aligned} 2x + 2 &= x+4 \\ \textcolor{red}{x} &= \textcolor{red}{2} \end{aligned}

Step 2: Solve for negative values:

\begin{aligned} -(2x + 2) &= x+4 \\ 3x &= -6 \\ \textcolor{red}{x} &= \textcolor{red}{-2} \end{aligned}

Step 3: Combine the solutions:

The solutions are \textcolor{red}{x = 2}  and  \textcolor{red}{x = -2}

For equations of the form |f(x)| = |g(x)|, it is easier to do solve them algebraically, using the following rule:

“If |a| = |b|, then a^2 = b^2 ”

So if |f(x)| = |g(x)|, then [f(x)]^2 = [g(x)]^2

Example: Solve |x-1| = |2x+3|

Step 1: Square both sides:

\begin{aligned} |x-1| &= |2x+3| \\ (x-1)^2 &= (2x+3)^2 \end{aligned}

Step 2: Expand and simplify:

\begin{aligned} x^2 - 2x + 1 &= 4x^2 + 12x + 9 \\ 3x^2 + 14x + 8 &= 0 \\ (3x+2)(x+4) &= 0 \end{aligned}

Step 3: So, the solutions are:

\textcolor{red}{x = - \dfrac{2}{3}}  and  \textcolor{red}{x = -4}