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
