Let f :[4, ∞) arrow[1, ∞) be a function defined by f ( x )=5 x ( x -4), then f -1( x ) is

Question

Let f :[4, ∞) arrow[1, ∞) be a function defined by f ( x )=5 x ( x -4), then f -1( x ) is (A) 2-√4+ log 5 x  (B) 2+√4+ log 5 x  (C) ((1/5)) x ( x -4) (D) None of these

Tardigrade Admin · Accepted Answer

Let y=5x(x-4) ⇒ x(x-4)= log 5 y ⇒ x2-4 x- log 5 y=0 ⇒ x=(4 ± √16+4 log 5 y/2) =(2 ± √4+ log 5 y) But x ≥ 4, so x=(2+√4+ log 5 y) ∴ f -1( x )=2+√4+ log 5 x

Q. Let $f : [4, \infty) \to [1, \infty)$ be a function defined by $f (x) = 5^{x (x - 4)}$ , then $f^{- 1} (x)$ is

$2 - 4 + lo g_{5} x$

$2 + 4 + lo g_{5} x$

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

None of these

Let
$y = 5^{x (x - 4)}$
$\Rightarrow x (x - 4) = lo g_{5} y$
$\Rightarrow x^{2} - 4 x - lo g_{5} y = 0$
$\Rightarrow x = \frac{4 \pm 16 + 4 l o g _{5} y}{2}$
$= (2 \pm 4 + lo g_{5} y)$
But $x \geq 4$ ,
so $x = (2 + 4 + lo g_{5} y)$
$∴ f^{- 1} (x) = 2 + 4 + lo g_{5} x$

Q. Let f:[4,∞)→[1,∞) be a function defined by f(x)=5x(x−4), then f−1(x) is

2−4+log5​x​

2+4+log5​x​

(51​)x(x−4)