Question

Find the domain and range of the following function given by

𝑓(𝑥)=√(3𝑥−5)(𝑥+4)/𝑥3−16𝑥

Answer 1

$$f(x)=\dfrac{\sqrt{(3x-5)(x+4)}}{x^3-16x}$$

$$(3x-5)(x+4)\geq0=>x\leq-4\ or\ x\geq\dfrac{5}{3}$$

$$x^3-16x\not=0=>x\not=-4, x\not=0, x\not=4$$

$$Domain: (-\infin, -4)\cup[\dfrac{5}{3}, 4)\cup(4, \infin)$$

$x\in (-\infin, -4), f(x)\in(-\infin, 0)$

$x\in [\dfrac{5}{3}, 4), f(x)\in(-\infin, 0]$

$x\in (4, \infin), f(x)\in(0,\infin)$

$$Range:(-\infin, \infin)$$