First slide

Functions (XII)

Question

Let f : [4, ∞) → [4, ∞) be a function defined by, f (x) = 5^{x(x – 4)}, then f^–1 (x) is

Moderate

A

$2 - \sqrt{4 + \log_{5} x}$
B

$2 + \sqrt{4 + \log_{5} x}$
C

${(\frac{1}{5})}^{x (x - 4)}$
D

None of these

Solution

Let y = 5^x(x–4) ⇒ x (x – 4) = log₅ y
$\begin{array}{l} \Rightarrow x^{2} - 4 x - \log_{5} y = 0 \\ \begin{aligned} \Rightarrow x & = \frac{4 \pm \sqrt{16 + 4 \log_{5} y}}{2} \\ = (2 \pm \sqrt{4 + \log_{5} y}) \end{aligned} \end{array}$
$\begin{array}{l} But x \geq 4, so x = (2 + \sqrt{4 + \log_{5} y}) \\ ∴ f^{- 1} (y) = 2 + \sqrt{4 + \log_{5} y} \end{array}$

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