Algebra of Real Functions

If $f\left(x\right)=\cos \left(\log x\right)$, find the value of $f\left(x\right).f\left(y\right)-\frac{1}{2}\left\{f\left(\frac{x}{y}\right)+f\left(xy\right)\right\}$

If $f\left(x\right)=\frac{4^x}{4^x+2}$, prove that $f\left(x\right)+f\left(1-x\right)=1$ and hence show that $f\left(\frac{1}{1997}\right)+f\left(\frac{2}{1997}\right)+f\left(\frac{3}{1997}\right)+ ....... +f\left(\frac{1996}{1997}\right)=998$

If $2f\left(x-1\right)-f\left(\frac{1-x}{x}\right)=x$, find the formula for $f\left(x\right)$.

If $f\left(x\right)=\log \left(x+\sqrt{1+x^2}\right)$, show that $f\left(-x\right)=-f\left(x\right)$.

If $f\left(x\right)=3x^2-6x+4$, for what values of $x$ is $3f\left(x\right)=f\left(3x\right)$?

If $f\left(x\right)=a{x}^2+bx+c$ and $f\left(1\right)=6, f\left(2\right)=11 \mathrm{and} f\left(3\right)=18$, find the values of $a, b \mathrm{and} c$ and hence find $f\left(-1\right)$.

If $f\left(x\right)=b.\frac{x-a}{b-a}+a.\frac{x-b}{a-b}$, prove that $f\left(a\right)+f\left(b\right)=f\left(a+b\right)$

If $f\left(x\right)=e^{ax+b}$, prove that $f\left(l\right).f\left(m\right).f\left(n\right)=f\left(l+m+n\right)e^{2b}$

Show that the following function is bounded and find the upper and lower bounds: $\tan x, -\frac{\mathrm{\pi }}{3}\le x\le \frac{\mathrm{\pi }}{3}$

Show that the following function is bounded and find the upper and lower bounds: $\sin x, -\mathrm{\pi }\le x\le \mathrm{\pi }$

Show that the following function is bounded and find the upper and lower bounds: $2x^2-3x+4, 1\le x\le 5$

Show that the following function is bounded and find the upper and lower bounds: $x^2+2x-3, 1\le x\le 4$

Show that $\cos x$ is a decreasing function of $x$ in $0<x<\frac{\mathrm{\pi }}{2}$.

Show that $\sin x$ is an increasing function of $x$ in $0<x<\frac{\mathrm{\pi }}{2}$.

If $f\left(x\right)={\sin }^{-1}x$, show that $\sin f\left(x\right)+\sin f\left(y\right)=\sin f\left(x+y\right)$.

If $f\left(x\right)=\tan x$, show that $f\left(x+y\right)=\frac{f\left(x\right)+f\left(y\right)}{1-f\left(x\right).f\left(y\right)}$.

If $f\left(x\right)=\frac{3x+2}{3x-2}$, prove that $\frac{f\left(x\right)+1}{f\left(x\right)-1}=\frac{3x}{2}$.

If $y=f\left(x\right)=\frac{x-1}{x-2}$, show that $f\left(\frac{1}{x}\right)=\frac{f\left(x\right)}{3f\left(x\right)-1}$.

If $f\left(x\right)=\frac{x-1}{x+1}$, prove that $\frac{f\left(x\right)-f\left(y\right)}{1+f\left(x\right).f\left(y\right)}=\frac{x-y}{1+xy}$.

If $f\left(x\right)=\frac{x}{1-x}$, show that $\frac{f\left(x+h\right)-f\left(x\right)}{h}=\frac{1}{\left(1-x\right){\displaystyle \left(1-x-h\right)}}$.