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Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsFunctions3 Marks
Question
If $\text{f(x)}=(\text{a}-\text{x}^\text{n})^{\frac{1}{\text{n}}},\text{a}>0$ and $\text{n}\in\text{N},$ then prove that f(f(x)) = x for all x.
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Answer

Given,
$\text{f(x)}=(\text{a}-\text{x}^\text{n})^{\frac{1}{\text{n}}},\text{a}>0$
Now,
$\text{f}(\text{f(x)})=\text{f}(\text{a}-\text{x}^\text{n})^{\frac{1}{\text{n}}}$
$=\bigg[\text{a}-\Big\{\big(\text{a}-\text{x}^{\text{n}}\big)^{\frac{1}{\text{n}}}\Big\}^{\text{n}}\bigg]^{\frac{1}{\text{n}}}$
$=\big[\text{a}-\big(\text{a}-\text{x}^{\text{n}}\big)\big]^{\frac{1}{\text{n}}}$
$=\big[\text{a}-\text{a}+\text{x}^\text{n}\big]^{\frac{1}{\text{n}}}$
$=(\text{x}^\text{n})^\frac{1}{\text{n}}$
$=(\text{x})^{\text{n}\times\frac{1}{\text{n}}}$
$=\text{x}$
$\therefore\ \text{f}(\text{f(x)})=\text{x}$ Hence, proved.

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