Let f(x) = loge x and g(x) = (2x^4-2x^3-x^2+2x-1/2x^2-2x+1) then domain of f(g(x)) for x > 0 is

Question

Let \(f(x)=\log _e x\ and\ g(x)=\left(\frac{2 x^4-2 x^3-x^2+2 x-1}{2 x^2-2 x+1}\right),\) then domain of \(f(g(x)) \ for \ x>0\) is

(1) \((1, \infty)\)

(2) \((0, \infty)\)

(3) \(\left(\frac{1}{2}, \infty\right)\)

(4) (0, 1)

Answer 1

Correct option is (1) \((1, \infty)\) 

Clearly \(2 x^2-2 x+1>0 \forall x \in R\)  

also \(\pm 1\) are roots of equation 

\(2 x^4-2 x^3-x^2-1=0 \)    

\(\Rightarrow 2 x^4-2 x^3-x^2+2 x-1=\left(2 x^2-2 x+1\right)(x-1)(x+1)\)  

\(\Rightarrow g(x)=(x-1)(x+1) \)  

\(f(g(x))=\log _e\left(x^2-1\right) \Rightarrow\left(x^2-1\right)>0 \)   

\(\Rightarrow x \in(-\infty,-1) \cup(1, \infty)\)