Functions Question 1
Question 1 - 2024 (01 Feb Shift 1)
Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ and $\mathrm{g}: \mathbf{R} \rightarrow \mathbf{R}$ be defined as
$f(x)=\left{\begin{array}{cc}\log _{e} x & , x>0 \ e^{-x} & , \quad x \leq 0\end{array}\right.$ and
nan
(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
$g(f(x))=\left{\begin{array}{l}f(x), f(x) \geq 0 \ e^{f(x)}, f(x)<0\end{array}\right.$
$\mathrm{g}(\mathrm{f}(\mathrm{x}))=\left{\begin{array}{l}e^{-x},(-\infty, 0] \ e^{\ln x},(0,1) \ \ln x,[1, \infty)\end{array}\right.$
Graph of $g(f(x))$
$\mathrm{g}(\mathrm{f}(\mathrm{x})) \Rightarrow$ Many one into
