### Functions Question 1

#### Question 1 - 2024 (01 Feb Shift 1)

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as

$f(x)=\begin{array}{cc}\log _e x & , x>0 \\ e^{-x} & , \quad x \leq 0\end{array}$

(1) one-one but not onto

(2) neither one-one nor onto

(3) onto but not one-one

(4) both one-one and onto

## Show Answer

#### Answer (2)

#### Solution

$g(f(x))=\begin{array}{l}f(x), f(x) \geq 0 \\ e^{f(x)}, f(x)<0\end{array}$

$g(f(x))=\begin{array}{l}e^{-x},(-\infty, 0] \\ e^{\ln x},(0,1) \\ \ln x,[1, \infty)\end{array}$

Graph of $g(f(x))$

$g(f(x)) \Rightarrow$ Many one into