Question

Find the domain of following functions.f(x)=(1)/(sqrt((0.2)^(x^(2)+9x+5)-125))

Answer 1

given function is $\frac{1}{\sqrt{(0.2)^{(x^2+9x+5)}-125}}$

we have to find the domain of the given function.

for square root to be defined,

$(0.2)^{(x^2+9x+5)}-125$ > 0

⇒$\frac{1}{5^{(x^2+9x+5)}}$ > 5³

⇒$5^{-(x^2+9x+5)}$ > 5³

⇒-(x² + 9x + 5) > 3

⇒x² + 9x + 5 < -3

⇒x² + 9x + 8 < 0

⇒x² + 8x + x + 8 < 0

⇒(x + 8)(x + 1) < 0

⇒-8 < x < -1

therefore, domain of the function f(x) ∈ (-8, -1)