Let f:R → R and g:R → R be defined as f(x) = { {logex, x>0} {e^-x, x ≤ 0} and

Question

Let \(\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}\) and \(\mathrm{g}: \mathbf{R} \rightarrow \mathbf{R}\) be defined as

\(f(x)=\left\{\begin{array}{cc}\log_ex & , \quad x > 0 \\ e^{-x} & , \quad x\le0\end{array}\right.\) and

\(g(x)=\left\{\begin{array}{cc}x & , \quad x \geq 0 \\ e^{x} & , \quad x<0\end{array}\right.\)

Then, gof: \(\mathbf{R} \rightarrow \mathbf{R}\) is :

(1) one-one but not onto

(2) neither one-one nor onto

(3) onto but not one-one

(4) both one-one and onto

Answer 1

Correct option is (2) neither one-one nor onto

\(g(f(x))=\left\{\begin{array}{l}f(x),& f(x) \geq 0 \\ e^{f(x)},& f(x)<0\end{array}\right.\)

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

Graph of \(g(f(x)) \)

\(g(f(x))\) ⇒ Many one into