Question
Find the domain of $$f\left( x \right) = \sqrt {{{\left( {0.625} \right)}^{4 - 3x}} - {{\left( {1.6} \right)}^{x\left( {x + 8} \right)}}} $$
A.
$$\left[ { - 3,\,2} \right]$$
B.
$$\left[ { 1,\,4} \right]$$
C.
$$\left[ {2,\,5} \right]$$
D.
$$\left[ { - 4,\, - 1} \right]$$
Answer :
$$\left[ { - 4,\, - 1} \right]$$
Solution :
$$\eqalign{
& {\text{Clearly, }}{\left( {0.625} \right)^{4 - 3x}} \geqslant {\left( {1.6} \right)^{x\left( {x + 8} \right)}} \cr
& {\text{or }}{\left( {\frac{5}{8}} \right)^{4 - 3x}} \geqslant {\left( {\frac{8}{5}} \right)^{x\left( {x + 8} \right)}} \cr
& {\text{or }}{\left( {\frac{8}{5}} \right)^{3x - 4}} \geqslant {\left( {\frac{8}{5}} \right)^{x\left( {x + 8} \right)}} \cr
& {\text{or }}3x - 4 \geqslant {x^2} + 8x \cr
& {\text{or }}{x^2} + 5x + 4 \leqslant 0 \cr
& {\text{or }} - 4 \leqslant x \leqslant - 1 \cr} $$
Hence, the domain of function $$f\left( x \right)$$ is $$x\, \in \left[ { - 4,\, - 1} \right]$$