The modulus function

In a nutshell

The modulus of a number is its non-negative numerical value. The function inside the modulus is called the argument of the modulus. When a whole function becomes the argument of the modulus, all outputs from that function become positive numbers on the $y$ -axis.

Example 1

A function is given as $f(x)=2x$ . Sketch $y=f(x)$ and $y=|f(x)|$ on the same set of axes and state their domains and ranges.

The domain of both functions is $x\in \R$ , but the range is different. For $f(x)$ it is $\R$ but for $\vert f(x)\vert$ it is $[0,+\infty)$ because the modulus can only span zero and positive numbers.

The sketch should look like this:

Maths; Functions; KS5 Year 13; The modulus function

The modulus function

A modulus function is a function of the type $y=\vert f(x)\vert$ :

If $f(x)\ge0\implies \vert f(x)\vert=f(x)$

If $f(x)\lt0\implies\vert f(x)\vert=-f(x)$

The domain of $\vert f(x)\vert$ and $f(x)$ is the same, but the range may change. All values of this function must be above the $x$ -axis.

Example 2

Sketch $f(x)=x^2+2x-3$ and $f(x)=\vert x^2+2x-3\vert$ .

Maths; Functions; KS5 Year 13; The modulus function

As you can see, both functions give the same coordinates when $f(x)$ is above the $x$ -axis, however $\vert f(x)\vert$ is a reflection of $f(x)$ when $f(x)$ is below the $x$ -axis.