Final answer:
The domain of the function f(x) = \sqrt{x^2 - 4} - 3 consists of all real numbers x such that x is greater than or equal to 2 or less than or equal to -2. This is because the square root requires the radicand to be non-negative. The domain in interval notation is (-\infty, -2] \cup [2, \infty).
Step-by-step explanation:
Determining the Domain of a Function
To determine the domain of the function f(x) = \sqrt{x^2 - 4} - 3, we need to consider the values of x for which the expression under the square root, x^2 - 4, is non-negative because the square root of a negative number is not defined in the set of real numbers. Therefore, we look for all x such that:
x^2 - 4 \geq 0
Rearranging, we get:
x^2 \geq 4
Which means:
x \geq 2 or x \leq -2
So, the domain of the function is all real numbers greater than or equal to 2, and all real numbers less than or equal to -2. In interval notation, this is written as (-\infty, -2] \cup [2, \infty).
When working with quadratic functions, one can often find themselves needing to solve for x by rearranging terms and employing the quadratic formula when needed. For instance:
ax^2 + bx + c = 0
Where the roots or solutions can be found using the formula:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
We would use such a process if we had to solve x^2 - 4 = 0 which is similar to the function's inner expression, but since we are interested in the domain, our goal is simply to understand the values of x for which the square root expression is valid.