Sets, Relations & Functions
Range of Composite Function
nta_pyq_2024_jan
Grade 11
If $f(x)=\begin{cases}2+2x,&-1\leq x<0\\1-\dfrac{x}{3},&0\leq x\leq3\end{cases}$; $g(x)=\begin{cases}-x,&-3\leq x\leq0\\x,&0<x\leq1\end{cases}$, then range of $(f\circ g)(x)$ is
Step-by-Step Solution
Key Concept: Determine the range of $g$, then apply $f$ to that range. For $x\in[-3,0]$: $g(x)=-x\in[0,3]$. For $x\in(0,1]$: $g(x)=x\in(0,1]$. Combined range of $g$: $[0,3]$. Apply $f$ to $[0,3]$: $f(t)=1-t/3$ for $t\in[0,3]$, giving $[0,1]$.
For $x\in[-3,0]$: $g(x)=-x\in[0,3]$; $f(g(x))=1-g(x)/3\in[0,1]$. For $x\in(0,1]$: $g(x)=x\in(0,1]$; $f(g(x))=1-x/3\in[2/3,1)$. Overall range $=[0,1]$.
Correct Answer: 3