Find the Domain of Definition of f(x) = cos-1(x² − 4)
Solution:
Given function:
\[ f(x) = \cos^{-1}(x^2 – 4) \]
For the inverse cosine function, the input must lie in the interval:
\[ -1 \leq x^2 – 4 \leq 1 \]
Add 4 to all parts:
\[ 3 \leq x^2 \leq 5 \]
Taking square root:
\[ \sqrt{3} \leq |x| \leq \sqrt{5} \]
So, the values of x are:
\[ x \in [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}] \]
Final Answer:
Domain = \[ [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}] \]