Let A= x :x 1 . The inverse of the function f : A A given by f(x)… - Vidyadip

MCQ

Let $\text{A}=\{\text{x}\in\text{R}:\text{x}\geq1\}.$ The inverse of the function $f : A \rightarrow A$ given by $\text{f(x)}=2^{\text{x}(\text{x}-1)},$ is:

A
$\big(\frac{1}{2}\big)^{\text{x}(\text{x}-1)}$
✓
$\frac{1}{2}\big\{1+\sqrt{1+4\log_2\text{x}}\big\}$
C
$\frac{1}{2}\big\{1-\sqrt{1+4\log_2\text{x}}\big\}$
D
$\text{Not defined}$

✓

Answer

Correct option: B.

$\frac{1}{2}\big\{1+\sqrt{1+4\log_2\text{x}}\big\}$

Given function is $\text{A}=\{\text{x}\in\text{R}:\text{x}\geq1\}.$
The inverse of the function $f : A \rightarrow A$ given by $\text{f(x)}=2^{\text{x}(\text{x}-1)}$
$\text{f(x)}=\text{y}$
$2^{\text{x}(\text{x}-1)}=\text{y}$
$\text{x}(\text{x}-1)=\log_2\text{y}$
$\text{x}^2+\text{x}=\log_2\text{y}$
$\text{x}^2+\text{x}+\frac{1}{4}=\log_2\text{y}+\frac{1}{4}$
$\Big(\text{x}-\frac{1}{2}\Big)^2=\frac{4\log_2\text{y}+1}{4}$
$\text{x}-\frac{1}{2}=\pm\sqrt{\frac{4\log_2\text{y}+1}{4}}$
$\text{x}=\frac{1}{2}\pm\sqrt{\frac{4\log_2\text{y}+1}{4}}$
$\text{x}=\frac{1}{2}+\sqrt{\frac{4\log_2\text{y}+1}{4}}$
$\text{f}^{-1}(\text{x})=\frac{1+\sqrt{4\log_2\text{y}+1}}{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Functions→Functions — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $y=e^{m \sin ^{-1} x}$ and $\left(1-x^2\right)\left(\frac{d y}{d x}\right)^2=A y^2$, then $A=$
$\qquad$

Let $\text{f(x)=}\begin{cases}\frac{\text{x}^4-5\text{x}^2+4}{|(\text{x}-1)(\text{x-2})|},\text{x}\neq1,2\\6, \text{x}=1\\12,\text{x}=2\end{cases}$ Then $f(x)$ is continuous on the set:

The integral value of $\int_{-2}^0\left[x^3+3 x^2+3 x+3+(x+1) \cos (x+1)\right] d x$ is

The direction ratios of the line which is perpendicular to the two lines
$\frac{x-7}{2}=\frac{y+17}{-3}=\frac{z-6}{1}$ and $\frac{x+5}{1}=\frac{y+3}{2}=\frac{z-6}{-2}$ are

The direction ratios of the line perprndicular to the lines $\frac{\text{x}-7}{2}=\frac{\text{y}+17}{-3}=\frac{\text{z}-6}{1}$ and, $\frac{\text{x}+5}{1}=\frac{\text{y}+3}{2}=\frac{\text{z}-4}{-2}$ are proportional to:

If $S = [S_{ij}]$ is a scalar matrix such that $S_{ij} = k$ and $A$ is a square matrix of the same order, then $AS = SA = ?$

The line joining the points (-2, 1, -8) and (a, b, c) is parallel to the line whose direction ratios are 6, 2, 3. The value of a, b, c are

Let A be an invertible matrix. Which of the following is not true?

The equation $\sin x+\cos x=2$ has

If $\cos \alpha, \cos \beta, \cos \gamma$ are the direction cosines of a line, then the value of $\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma$ is