Question
Let $$f\left( x \right)$$ be a function whose domain is $$\left[ { - 5,\,7} \right].$$ Let $$g\left( x \right) = \left| {2x + 5} \right|.$$ Then the domain of $$\left( {f\,o\,g} \right)\left( x \right)$$ is :
A.
$$\left[ { - 5,\,1} \right]$$
B.
$$\left[ { - 4,\,0} \right]$$
C.
$$\left[ { - 6,\,1} \right]$$
D.
none of these
Answer :
$$\left[ { - 6,\,1} \right]$$
Solution :
$$\eqalign{
& f\left\{ {g\left( x \right)} \right\} = f\left( {\left| {2x + 5} \right|} \right)\,\,\,\,\,\,\,\,\,\therefore \left| {2x + 5} \right| \in \left[ { - 5,\,7} \right] \cr
& \therefore - 5 \leqslant \left| {2x + 5} \right| \leqslant 7 \cr
& \Rightarrow \left| {2x + 5} \right| \leqslant 7 \cr
& \Rightarrow - 7 \leqslant 2x + 5 \leqslant 7 \cr
& \Rightarrow - 6 \leqslant x \leqslant 1 \cr} $$