Sets, Relations & Functions
Composition of Functions — One-One and Onto
nta_pyq_2024_jan
Grade 11
Question:
Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=\begin{cases}\log_e x,&x>0\\e^{-x},&x\leq0\end{cases}$ and $g(x)=\begin{cases}x,&x\geq0\\e^x,&x<0\end{cases}$. Then $g\circ f:\mathbb{R}\to\mathbb{R}$ is:
one-one but not onto
neither one-one nor onto
onto but not one-one
both one-one and onto
Step-by-Step Solution
Key Concept: Compute $g(f(x))$ for the three intervals $x\leq0$, $0<x<1$, $x\geq1$. Check injectivity (one-one) and surjectivity (onto) by analyzing the range and whether two inputs give the same output.
$g(f(x))=\begin{cases}e^{-x},&x\leq0\\x,&0<x<1\\\ln x,&x\geq1\end{cases}$. $g(f(0))=1$ and $g(f(1))=1$: many-one. Range $=(-\infty,0)\cup[1,\infty)$: not onto. Neither one-one nor onto.
Correct Answer: 2