Amit M Agarwal Solutions for Chapter: Functions, Exercise 9: Proficiency in 'Functions' Exercise 1

Let $f:R\to R$ be a function defined by $f\left(x+1\right)=\frac{f\left(x\right)-5}{f\left(x\right)-3}\forall x\in R$. Then, which of the following statements is/are true?

Let $f\left(x\right)=1-x-x^3$. Then, the real values of $x$ satisfying the inequality, $1-f\left(x\right)-f^3\left(x\right)>f\left(1-5x\right)$, are

If a function satisfies $\left(x-y\right)f\left(x+y\right)-\left(x+y\right)f\left(x-y\right)=2\left(x^{2}y-y^3\right), \forall x, y\in R$ and $f\left(1\right)=2,$ then

If the fundamental period of function $f\left(x\right)=\mathrm{s}\mathrm{i}\mathrm{n}x+\mathrm{c}\mathrm{o}\mathrm{s}\left(\sqrt{4-a^2}\right)x$ is $4\pi$, then the value of $a$ is/are:

Let $f\left(x\right)$ be a real-valued function such that $f\left(0\right)=\frac{1}{2}$ and $f\left(x+y\right)=f\left(x\right)f\left(a-y\right)+f\left(y\right)f\left(a-x\right) \forall x, y\in R$, then for some real $a$:

If $f\left(g\left(x\right)\right)$ is one-one function, then:

Which of the following functions have their range equal to $R$ (the set of real numbers)?

Which of the following pairs of functions are identical?