Question

The function $f:R \rightarrow R$ defined by $f(x) = e^{x}$ is

• A. Onto
• B. Many-one
• C. One-one and into
• D. Many one and onto

Answer 1

Answer: (C)

One-one and into

Answer 2

Function $f:R \rightarrow R$ is defined by $f(x) = e^{x}$. Let

$x_{1},x_{2} \in R$ and $f(x_{1}) = f(x_{2})$ or $e^{x_{1}} = e^{x_{2}}$ or $x_{1} = x_{2}$. Therefore f is one-one. Let $f(x) = e^{x} = y$. Taking log on both sides, we get $x = \log y$. We know that negative real numbers have no pre-image or the function is not onto and zero is not the image of any real number. Therefore function f is into.