Q111 / 200 · AP EAMCET 2001 · Mathematics

$f(x)=\left(20-x^4\right)^{1 / 4} \quad$ for $\quad 0 < x < \sqrt{5}$, then $f\left(f\left(\frac{1}{2}\right)\right)$ is equal to

  • A
    $2^{-4}$
  • B
    $2^{-3}$
  • C
    $2^{-2}$
  • Correct answer
    D
    $2^{-1}$

Solution

We have, $\begin{aligned} f(x) & =\left(20-x^4\right)^{\frac{1}{4}} \\ f\left(\frac{1}{2}\right) & =\left(20-\frac{1}{16}\right)^{\frac{1}{4}}=\left(\frac{319}{16}\right)^{\frac{1}{4}} \\ f\left[f\left(\frac{1}{2}\right)\right] & =f\left[\left(\frac{319}{16}\right)^{\frac{1}{4}}\right] \\ & =\left(20-\frac{319}{16}\right)^{\frac{1}{4}} \\ & =\left(\frac{1}{16}\right)^{\frac{1}{4}}=\frac{1}{2}=2^{-1}\end{aligned}$

Hence, the correct answer is option: 'D'.

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