First slide

Functions (XII)

Question

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

Moderate

A

$(- 4, 1)$
B

$(- 4, - 1)$
C

$[- 4, 1]$
D

$[- 4, - 1]$

Solution

$\begin{array}{l} Clearly, (0.625)^{4 - 3 x} \geq (1.6)^{x (x + 8)} \\ ⇒ {(\frac{5}{8})}^{4 - 3 x} \geq {(\frac{8}{5})}^{x (x + 8)} \\ ⇒ {(\frac{8}{5})}^{3 x - 4} \geq {(\frac{8}{5})}^{x (x + 8)} \\ ⇒ 3 x - 4 \geq x^{2} + 8 x or x^{2} + 5 x + 4 \leq 0 \\ ⇒ (x + 1) (x + 4) \leq 0 \\ ⇒ - 4 \leq x \leq - 1 \\ Hence, the domain of function f (x) is x \in [- 4, - 1] . \end{array}$

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