If the function f -1- - -1- - is defined by f x -2 x x -1- then f -1 x isA- 1-21-1-4 log 2 xB- 1-2xx-10C- 1-21-1-4 log 2 xD- Not defined

The correct option is A Given f(x)=2^x(x 1) ⇒ x(x 1)= log_2f(x) ⇒ x^2 x log_2f(x)=0⇒ x=1±√(1+4log_2f(x))/2 Onlyx=1+√(1+4log_2f(x))/2 lies in the domain ∴ f^ 1 ...

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Byju's Answer

Standard XII

Mathematics

Quadratic Formula for Finding Roots

If the functi...

Question

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

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Not defined

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Solution

The correct option is B

Given $f (x) = 2^{x (x - 1)} \Rightarrow x (x - 1) = l o g_{2} f (x)$

$\Rightarrow x^{2} - x - l o g_{2} f (x) = 0 \Rightarrow x = \frac{1 \pm \sqrt{1 + 4 l o g_{2} f (x)}}{2}$

Only $x = \frac{1 + \sqrt{1 + 4 l o g_{2} f (x)}}{2}$ lies in the domain

$∴ f^{- 1} (x) = \frac{1}{2} [1 + \sqrt{1 + 4 l o g_{2} x}]$

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Similar questions

Q. If the function

f : [1, \infty) \to [1, \infty)

is defined by

f (x) = 2^{x (x - 1)}

, then

f^{- 1} (x)

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f : [1, \infty) \to [1, \infty)

is defined by

f (x) = 2^{x (x - 1)}

, then

f^{- 1} (x)

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f : [1, \infty) \to [1, \infty)

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Q. Let

A = \{x \in R : x \geq 1\}

. The inverse of the function,

f : A \to A

given by

f (x) = 2^{x (x - 1)}, is

(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) not defined

f : (0, \infty) \to (0, \infty)

defined by

f (x) = {\begin{matrix} 2^{x} & x \in (0, 1) 5^{x} & x \in [1, \infty) \end{matrix}

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If the function f:(1,∞)→(1,∞) is defined by f(x)=2x(x−1) , then f−1(x) is

The correct option is B Given f(x)=2x(x−1) ⇒ x(x−1)= log2f(x) ⇒x2−x−log2f(x)=0⇒ x=1±√1+4log2f(x)2 Onlyx=1+√1+4log2f(x)2 lies in the domain ∴ f−1(x)=12[1+√1+4log2x]

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