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The modulus function

The modulus function

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Explainer Video

Tutor: Alice

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 yyy​-axis.


Example 1

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


The domain of both functions is x∈Rx\in \Rx∈R, but the range is different. For f(x)f(x)f(x) it is R\RR but for ∣f(x)∣\vert f(x)\vert∣f(x)∣ it is [0,+∞)[0,+\infty)[0,+∞) 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)\verty=∣f(x)∣ :

 ​​

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

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


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


Example 2

Sketch f(x)=x2+2x−3f(x)=x^2+2x-3f(x)=x2+2x−3 and f(x)=∣x2+2x−3∣f(x)=\vert x^2+2x-3\vertf(x)=∣x2+2x−3∣.

Maths; Functions; KS5 Year 13; The modulus function

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


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FAQs - Frequently Asked Questions

What happens to the range of a modulus function?

The range is restricted to zero and positive numerical values.

What happens to the domain of a modulus function?

It is the same as the original function.

What is the modulus function?

The modulus function, or absolute value, is simply a number without the sign. The results is always a non-negative numerical value.

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