Math  /  Algebra

QuestionWhat is the domain of the function f(x)=x281f(x)=\sqrt{x^{2}-81} ?
The domain is \square (Type your answer in interval notation.)

Studdy Solution

STEP 1

1. The function f(x)=x281 f(x) = \sqrt{x^2 - 81} is defined for real numbers.
2. The expression under the square root, x281 x^2 - 81 , must be non-negative for f(x) f(x) to be real-valued.

STEP 2

1. Set up the inequality for the expression inside the square root.
2. Solve the inequality to find the values of x x .
3. Express the solution in interval notation.

STEP 3

To ensure the square root is defined, set up the inequality: x2810 x^2 - 81 \geq 0

STEP 4

Rearrange the inequality: x281 x^2 \geq 81

STEP 5

Solve the inequality x281 x^2 \geq 81 by taking the square root of both sides: x9 |x| \geq 9
This implies: x9orx9 x \leq -9 \quad \text{or} \quad x \geq 9

STEP 6

Express the solution in interval notation: The domain of f(x) f(x) is: (,9][9,) (-\infty, -9] \cup [9, \infty)
The domain is (,9][9,)(- \infty, -9] \cup [9, \infty).

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