The function f: R → R defined by f(x)=2^x+2^|x|, is

The correct answer is Option (3) → one-one and into

Clearly, $f(x)=2^x+2^{|x|}> 0$ for all $x ∈ R$.

So, Range (f) ≠ R (Co-domain of f).

∴ f is an into function.

We observe that for any $x, y ∈ R$

$x≠y⇒2^x≠2^y=2^x+2^{|x|}≠2^y+2^{|y|}⇒ f(x) ≠ f (y)$

∴ f is a one-one function.

Hence, f is one-one and into.