Inverse Function

The function $\displaystyle f:R\to R$ is defined by $\displaystyle f(x)={{4}^{x}}-2$ .

$\displaystyle (a)$ Find the value of $\displaystyle x$ for which $\displaystyle f(x)=0$ .

$\displaystyle (b)\ $ Find the inverse function $\displaystyle {{f}^{{-1}}}$ , and state the domain of $\displaystyle {{f}^{{-1}}}$ .


Solution

$\displaystyle \ \ \ \ \ \ \ \ \ \ f(x)={{4}^{x}}-2$

$\displaystyle (a)\ \ \ \ \ f(x)=\ 0$

$\displaystyle \ \ \ \ \ \ \ \ {{4}^{x}}-2=0$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ {{4}^{x}}=2$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ {{2}^{{2x}}}=2$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ 2x=1$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=\frac{1}{2}$

$\displaystyle (b)\ \ \ \ Let\ \ {{f}^{{-1}}}(x)=y$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=f(y)$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x={{4}^{y}}-2$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{4}^{y}}=x+2$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y={{\log }_{4}}(x+2)$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{f}^{{-1}}}(x)={{\log }_{4}}(x+2)\ ,\ x>-2$

$\displaystyle \text{The domain of  }{{f}^{{-1}}}=\{x/x\in R,x>-2\}$