First slide

Functions (XII)

Question

If the function $f : [1, \infty) \to [1, \infty)$ is defined by $f (x) = 2^{x (x - 1)} then f^{- 1} (x)$ is

Moderate

A

${(\frac{1}{2})}^{x (x - 1)}$
B

$\frac{1}{2} (1 + \sqrt{1 + 4 \log_{2} x})$
C

$\frac{1}{2} (1 - \sqrt{1 + 4 \log_{2} x})$
D

$\frac{1}{2} (1 \pm \sqrt{1 + 4 \log_{2} x})$

Solution

$y = 2^{x (y - 1)} \Rightarrow x (x - 1) = \log_{2} y$
$\Rightarrow x^{2} - x - \log_{2} y = 0$
$∴ x = \frac{1 \pm \sqrt{1 + 4 \log_{2} y}}{2}$
$∴ f^{- 1} (x) = \frac{1}{2} \{1 + \sqrt{1 + 4 \log_{2} y}\} [∵ x \geq 1]$

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