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:

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The modulus function

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

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

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.