Embibe Experts Solutions for Chapter: Functions, Exercise 2: Level 2

Let $f$ be a function defined on $\left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$ $b$ $f\left(x\right)=3{\cos }^{4}x-6{\cos }^{3}x-6{\cos }^{2}x-3.$ Then, the range of $f\left(x\right)$ is

Let $f$ be a function with the following properties:
(i) $f\left(1\right)=1$, and
(ii) $f\left(2n\right)=n\times f\left(n\right)·$ for any positive integer $n$.
If the value of $f\left(2^{100}\right)=2^{49xy}$, then the digits $x$ and $y$ respectively?

The number of solutions of the equation $\left[x\right]+2\left\{-x\right\}=3x,$ is (where$\left[·\right]$ represents the greatest integer function and $\left\{x\right\}$ denotes the fractional part of $x$):

The function $f\left(x\right)={\sin }^4\left(x+\frac{\pi }{6}\right)+{\cos }^4\left(x+\frac{\pi }{6}\right)$ is

$f\left(x\right)=\sin x+\cos x,g\left(x\right)=x^2-1$ then $g\left(f\right(x\left)\right)$ is invertible if

Which of the following functions is inverse to itself?

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

Let $f:\left[-\frac{\pi }{3},\frac{2\pi }{3}\right]\to [0,4]$ be a function defined as $f\left(x\right)=\sqrt{3}\sin x-\cos x+2.$ Then $f^{-1}\left(x\right)$ is given by