Telangana Board Solutions for Exercise 1: Exercise 1 (a)

If $f:[0,\infty )\to [0,\infty )$ defined by $f\left(x\right)=x^2$ then $f$ is

If $f:\mathit{R}\to [0,\infty )$ defined by $f\left(x\right)=x^2$ then $f$ is

If $f:\mathit{R}\to \mathit{R}$ defined by $f\left(x\right)=x^2$ then $f$ is

Is $g=\left\{\left(1,1\right),\left(2,3\right),\left(3,5\right),\left(4,7\right)\right\}$ a function from $A=\left\{1,2,3,4\right\} \text{to} B=\left\{1,3,5,7\right\}?$ If this is given by the formula $g\left(x\right)=ax+b$, then find $a \text{and} b$.

If the function $f:\mathit{R}\to \mathit{R}$ defined by $f\left(x\right)=\frac{3^x+3^{-x}}{2}$, then show that $f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)f\left(y\right)$.

If the function $f:\mathit{R}\to \mathit{R}$ defined by $f\left(x\right)=\frac{4^x}{4^x+2}$, then show that $f\left(1-x\right)=1-f\left(x\right)$, and hence deduce the value of $f\left(\frac{1}{4}\right)+2f\left(\frac{1}{2}\right)+f\left(\frac{3}{4}\right)$.

If the function $f:\left\{-1,1\right\}\to \left\{0,2\right\}$, defined by $f\left(x\right)=ax+b$ is a surjection, then find $a \text{and} b$.

If $f\left(x\right)=\cos \left(\log x\right)$, then show that $f\left(\frac{1}{x}\right)·f\left(\frac{1}{y}\right)-\frac{1}{2}\left(f\left(\frac{x}{y}\right)+f\left(xy\right)\right)=0$.