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Q.
A function $g$ has domain $[0,2]$ and range $[-1,3]$. The domain and range of the function $f$ defined by $f(x)=3-2 g\left(9 x^2-1\right)$ are equal to
Relations and Functions - Part 2
Solution:
We have $f(x)=3-2 g\left(9 x^2-1\right)$
As domain of $g(x)=[0,2]$ so, for domain of $f(x)$,
$0 \leq 9 x ^2-1 \leq 2 \Rightarrow \frac{1}{9} \leq x ^2 \leq \frac{1}{3} \Rightarrow x \in\left[\frac{-1}{\sqrt{3}}, \frac{-1}{3}\right] \cup\left[\frac{1}{3}, \frac{1}{\sqrt{3}}\right]=\text { Domain of } f ( x ) \text {. }$
As range of $g ( x )=[-1,3]$
$\Rightarrow-1 \leq g \leq 3 \Rightarrow-6 \leq-2 g \leq 2 \Rightarrow-3 \leq 3-2 g \leq 5 \Rightarrow-3 \leq f ( x ) \leq 5$
So, range of $f ( x )=[-3,5]$.