Question

Find the range of the real valued function: f(x) = log |4 - x²|

Answer 1

we know, The $\textbf{domain}$ of the function $\text{f(x)}$ is the set of all possible values for which function is defined. and range of the function is the set of all values that $f(x)$ takes.

we have to find range of real valued of function : f(x) = log|4 - x²|

first of all, find domain of function f(x) = log|4 - x²|

To define f(x),
|4 - x²| > 0
it is correct for all real value of x except x = -2, 2

hence, domain of function $\in\mathbb{R}-\{-2,2\}$

$\implies$ domain of function $\in$ (-∞ , -2) U (-2, 2) U (2, ∞ )

now, range of function :

differentiate f(x) with respect to x,

f'(x) = -2x/(4 - x²) , for all

f'(x) = 0 => x = 0

f'(x) = undefined => x = ±2

so, find value of f(x) at x tends to -∞, ∞ and x = 0

we get, f(-∞) = ∞ , f(0) = log4 , f(2) = -∞ and f(-2) = -∞

hence, range of function $\in\mathbb{R}$