Algebra of Functions

The function $f\left(x\right)=0$ has eight distinct real solution and $f$ also satisfy $f\left(4+x\right)=f\left(4-x\right).$ The sum of all the eight solutions of $f\left(x\right)=0$ is –

If $f\left(x\right)+2f(1-x)=x^2+1, \forall x\in R$, then the range of $f$ is

If $g\left(x\right)=\frac{1}{2}\left(e^x+e^{-x}\right)$ and $f\left(x\right)=\frac{1}{2}\left(e^x-e^{-x}\right)$ then show that

[b] $g\left(x\right){]}^2-\left[f\right(x){]}^2=1$

If $g\left(x\right)=\frac{1}{2}\left(e^x+e^{-x}\right)$ and $f\left(x\right)=\frac{1}{2}\left(e^x-e^{-x}\right)$ then show that

[a] $g(x+y)=g\left(x\right) g\left(y\right)+f\left(x\right) f\left(y\right)$

If $f(x)=x+1, g(x)=x-2$, then solve the equation $|f(x)+g(x)|=|f(x)|+|g(x)|$.

If $f(x)=\sin (\log x)$ and $g(x)=\cos (\log x)$, then find the value of $f(x) \cdot g(y)-\frac{1}{2}\left[f(x y)+f\left(\frac{x}{y}\right)\right]$.

If $f(x)=3 \sin x$ and $\phi(x)=\sin ^{2} x$, then find the value of $(f+\phi)\left(\frac{\pi}{3}\right)$.

If $f(x)=x^{3}+2 x^{2}$ and $g(x)=3 x^{2}-1$, then find $f+g, f-g, f g$ and $\frac{f}{g}$ and state their domains.

Let $f$ be a function satisfying $f(x+y)=f(x)+f(y)$ for all real $x$ and $y$ and if $f(1)=k$, then show that $f(n)=n k $ for all +ve integer $n$.

If $f^2\left(x\right)·f\left(\frac{1-x}{1+x}\right)=x^3$, $x\notin \left\{-\mathrm{1,1}\right\}$ and $f\left(x\right)\ne 0$, then the value of $\left|\left[f\left(-2\right)\right]\right|$ is (where $\left[.\right]$ is the greatest integer function).

If $f^2\left(x\right)·f\left(\frac{1-x}{1+x}\right)=x^3, x\ne -\mathrm{1,} 1$ and $f\left(x\right)\ne 0$], then find $\left|\left[f\left(-2\right)\right]\right|$ (where $\left[.\right]$ is the greatest integer function).

Let $A$ denote the set of all real numbers $x$ such that $x^3-[x{]}^3=(x-\left[x\right]{)}^3,$ where $\left[x\right]$ is the greatest integer less than or equal to $x.$ Then

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x-1$, then find the value of $\left(f-g\right)\left(41\right)$.

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x+1$, then find the value of $\left(f+g\right)\left(59\right)$.

Number of functions $f:\left[0,1\right]\to \left[0,1\right]$ satisfying $\left|f\left(x\right)-f\left(y\right)\right|=\left|x-y\right|$ for all $x, y$ in $\left[0,1\right]$ is

Let $f\left(x\right)=\left\{\begin{array}{l}x\sin \left(\frac{1}{x}\right) , \text{when} x\ne 0\\ 1 , \text{when} x=0\end{array}\right\}$ and $A=\left\{x\in R:f\left(x\right)=1\right\}$. Then, $A$ has

Reduction of the proper fraction $\frac{f\left(x\right)}{g\left(x\right)}$ into a sum of partial fraction depends on the factorization of

Exactly how many functions $f:Q\to Q$ exist such that $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$ and$f\left(xy\right)=f\left(x\right)f\left(y\right)$ for all $x, y\in Q ?$

If $f:R\to R$ is defined as $f\left(x\right)=\frac{{2020}^x}{{2020}^x+\sqrt{2020}}, \forall x\in R,$ then $\sum _{r=1}^{4039}2f\left(\frac{r}{4040}\right)=$

If ${\left(f\left(x\right)\right)}^{2=}f\left(x^2\right)+f\left(1\right)$ holds good, then find $f\left(x\right)$