To determine the domain of the function \( f(x) = \sqrt{2x^2 - 50} \), we need to ensure that the expression inside the square root is non-negative (since we cannot take the square root of a negative number in the realm of real numbers).
Therefore, we need to solve the inequality:
\[ 2x^2 - 50 \geq 0 \]
First, let's isolate \( x^2 \):
\[ 2x^2 \geq 50 \]
\[ x^2 \geq 25 \]
Next, we take the square root of both sides, remembering to consider both the positive and negative roots:
\[ |x| \geq 5 \]
This gives us two inequalities:
\[ x \leq -5 \quad \text{or} \quad x \geq 5 \]
Thus, the domain of \( f(x) \) in interval notation is:
\[ (-\infty, -5] \cup [5, \infty) \]
This means \( f(x) \) is defined for all \( x \) such that \( x \) is less than or equal to -5 or greater than or equal to 5.
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