Question

f(x) is

• A. One-One onto
• B. One-One Into
• C. Many-One Onto
• D. Many-One Into

Answer 1

The correct option is D

Many-One Into

$f\left(x\right)=2^x+2^{|x|}$

For $x\ge 0$, $f\left(x\right)={2.2}^x$

And for $x<0,f\left(x\right)=2^x+2^{-x}$

In this case $f^{\prime }\left(x\right)=2^x\ln 2-2^{-x}\ln 2$ $=\ln 2⟮\frac{2^{2x}-1}{2^x}⟯$ $=\frac{\ln 2}{2^x}⟮2^x+1⟯⟮2^x-1⟯<0$

So, f(x) is decreasing in $x\in (-\infty ,0)$ and increasing in $x\in (0,\infty )$

So, f(x) is many one

Also, Range of f(x) is $[2,\infty )$

f is many-one Into