Question

The function $f:R\to R$ defined by $f\left(x\right)=6^x+6$ is

• A. one-one and onto
• B. many-one and onto
• C. one-one and into
• D. mauny-one and into

Answer 1

The correct option is A one-one and onto

$f:R\to R$

$f\left(x\right)=6^x+6$

$\text{For one - one,}$

$f\left(x_1\right)=f\left(x_2\right)$

$\Rightarrow 6^{x_1}+6$ &=$6^{x_2}+$

$\Rightarrow 6^{x_1}=6^{x_2}$

$\Rightarrow x_1=x_2$

Therefore f is one - one.

Now,

$\begin{array}{cc}f\left(x\right)& =y=6^x+6\\ & \Rightarrow (y-6)=6^x\end{array}$

Taking log both side.

$\Rightarrow {\log }_6(y-6)=x$

Now,

$f\left({\log }_6\right(y-6\left)\right)=6^{{\log }_6(y-6)}+6$

$=y-6+6$

$=y$

$\therefore f\text{is onto}$

Hence, f is one - one and onto.