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  4. Let f(x)=(1/2)- tan ((π x/2)),-1<x<1 and g(x)= √3+4 x-4 x2 then the domain (f+g) is

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Q. Let $f(x)=\frac{1}{2}-\tan \left(\frac{\pi x}{2}\right),-1
TS EAMCET 2021

Solution:

$ f(x)=\frac{1}{2}-\tan \left(\frac{\pi x}{2}\right),-1< x< 1 $
$g(x)=\sqrt{3+4 x-4 x^2}$
$ =\sqrt{4-(2 x-1)^2}$
Domain of $f(x) \Rightarrow \frac{\pi x}{2} \neq(2 n+1) \frac{\pi}{2}$
$x \neq 2 n+1$
Domain $\Rightarrow x \in(-1,1)$
Domain of $g(x) \Rightarrow 4-(2 x-1)^2 \geq 0$
$\Rightarrow(2 x-1)^2 \leq 4 $
$ \Rightarrow-2 \leq 2 x-1 \leq 2$
$\Rightarrow-\frac{1}{2} \leq x \leq \frac{3}{2}$
Domain of $g(x) \Rightarrow x \in\left[-\frac{1}{2}, \frac{3}{2}\right]$
Domain of $f(x)+g(x)$ will be
$ x \in(-1,1) \cap x \in\left[-\frac{1}{2}, \frac{3}{2}\right] $
$\Rightarrow x \in\left[-\frac{1}{2}, 1\right)$