Question

The domain of the function $f\left(x\right)=\sqrt{{0.625}^{4-3x}-{1.6}^{x(x+8)}}$ is

• A. $(-1,4)$
• B. $[-1,4]$
• C. $[-4,-1]$
• D. $(-4,-1)$

Answer 1

The correct option is C $[-4,-1]$

$f\left(x\right)=\sqrt{{0.625}^{4-3x}-{1.6}^{x(x+8)}}$
For $f\left(x\right)$ to be defined
${0.625}^{4-3x}-{1.6}^{x(x+8)}\ge 0$
$\Rightarrow {0.625}^{4-3x}\ge {1.6}^{x(x+8)}\Rightarrow {\left(\frac{5}{8}\right)}^{4-3x}\ge {\left(\frac{8}{5}\right)}^{x(x+8)}\Rightarrow {\left(\frac{8}{5}\right)}^{3x-4}\ge {\left(\frac{8}{5}\right)}^{x(x+8)}\Rightarrow 3x-4\ge x^2+8x\Rightarrow x^2+5x+4\le 0\Rightarrow (x+4)(x+1)\le 0\therefore x\in [-4,-1]$