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The modulus function

The modulus function

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Summary

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 yy​-axis.


Example 1

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


The domain of both functions is xRx\in \R, but the range is different. For f(x)f(x) it is R\R but for f(x)\vert f(x)\vert it is [0,+)[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=f(x)y=\vert f(x)\vert :

 ​​

If f(x)0 f(x)=f(x)f(x)\ge0\implies \vert f(x)\vert=f(x)​​

If f(x)<0 f(x)=f(x)f(x)\lt0\implies\vert f(x)\vert=-f(x)​​


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


Example 2

Sketch f(x)=x2+2x3f(x)=x^2+2x-3 and f(x)=x2+2x3f(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)f(x)​ is above the xx​-axis, however f(x)\vert f(x)\vert  is a reflection of  f(x)f(x)​ when f(x)f(x) is below the xx​-axis.


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Composite and inverse functions - Higher

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Composite and inverse functions - Higher

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Functions

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The modulus function

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The modulus function

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