If $f(x)=\frac{a^x+a^{-x}}{2}$ then $f(x+y)f(x-y)$ is equal to:

Find the domain and range of each of the following real value functions:
$f(x)=-|x|$

Let $f:R^{+}\to R$, where $R^{+}$ is the set of all positive real numbers, be such that $f(x)={\log }_{e}x$.
Determine
whether $f(xy)=f(x)+f(y)$ holds.

Let $f,g$ : $R\to R$ be defined respectively by $f(x)=x+1,g(x)=2x-3$. Find $f+g,f-g$ and $\frac{f}{g}$

If $$f:R\rightarrow R$$ and $$g:R\rightarrow R$$ are defined by $$f\left( x \right) =x-\left[ x \right]$$ and $$ g\left( x \right) =\left[ x \right]$$  for $$x\in R$$,where $$[x]$$ is the greatest integer not exceeding $$x$$,then for every  $$x\in R$$,$$f(g(x))$$ is equal to

If $f(n)=\frac{4n+3}{6n-4},n\ne \frac{2}{3}$ show that $f[f(n)]=n$

If $A=\left(\begin{array}{cc}\frac{-1+i\sqrt{3}}{2i}& \frac{-1-i\sqrt{3}}{2i}\\ \frac{1+i\sqrt{3}}{2i}& \frac{1-i\sqrt{3}}{2i}\end{array}\right),i=\sqrt{-1}$ and $f(x)=x^2+2$. Then $f(A)$ equals

If $\varnothing (x)={\varnothing }^{\prime }(x)$  and $\varnothing (1)=2,$ then $\varnothing (3)=?$

If $f(x)=\frac{x}{1-x}=\frac{1}{y}$, then the value of $f(y)$: