The function f : R → R defined by f (x) = ex is OntoMany - oneOne - one  and into Many one and onto

The function f : R → R defined by f (x) = e^x is both one-to-one and onto.

Here's why:

1. One-to-one (Injective): A function is said to be one-to-one (or injective) if the preimage of any two different elements in the codomain are different. In other words, if f(x1) = f(x2), then x1 must equal x2. For the function f(x) = e^x, if we assume that e^x1 = e^x2, taking the natural logarithm of both sides gives us x1 = x2. Therefore, the function is one-to-one.

2. Onto (Surjective): A function is said to be onto (or surjective) if every element in the codomain has a preimage in the domain. In other words, for any y in the codomain, there exists an x in the domain such that f(x) = y. For the function f(x) = e^x, given any y in R, we can find an x in R (specifically, x = ln(y)) such that e^x = y. Therefore, the function is onto.

So, the function f(x) = e^x is both one-to-one and onto.