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

one-one and onto many-one and onto one-one and into many-one and into

one-one and into

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.