Question
The Domain for which the function $$f\left( x \right) = 2{x^2} - 1$$ and $$g\left( x \right) = 1 - 3x$$ is equal, i.e., $$f\left( x \right) = g\left( x \right),$$ is :
A.
$$\left\{ {0,\,2} \right\}$$
B.
$$\left\{ {\frac{1}{2},\, - 2} \right\}$$
C.
$$\left\{ { - \frac{1}{2},\,2} \right\}$$
D.
$$\left\{ {\frac{1}{2},\,2} \right\}$$
Answer :
$$\left\{ {\frac{1}{2},\, - 2} \right\}$$
Solution :
$$\eqalign{
& {\text{For }}f\left( x \right) = g\left( x \right) \cr
& \Rightarrow 2{x^2} - 1 = 1 - 3x \cr
& \Rightarrow 2{x^2} + 3x - 2 = 0 \cr
& \Rightarrow 2{x^2} + 4x - x - 2 = 0 \cr
& \Rightarrow 2x\left( {x + 2} \right) - 1\left( {x + 2} \right) = 0 \cr
& \Rightarrow \left( {x + 2} \right)\left( {2x - 1} \right) = 0 \cr
& \Rightarrow x = - 2,\,\frac{1}{2} \cr} $$
$$\therefore $$ The domain for which the function $$f\left( x \right) = g\left( x \right){\text{ is }}\left\{ { - 2,\,\frac{1}{2}} \right\}$$