Math Problem Statement

Find the domain of the function h(x) = (√(x - 8)) / (x(x - 9)).

Solution

To find the domain of the function h(x)=x8x(x9)h(x) = \frac{\sqrt{x - 8}}{x(x - 9)}, we need to determine when the function is defined. There are two primary considerations:

  1. The square root condition: The expression under the square root, x8x - 8, must be non-negative (i.e., x80x - 8 \geq 0) because square roots of negative numbers are undefined in the real number system. This gives: x8x \geq 8

  2. The denominator condition: The denominator x(x9)x(x - 9) must not be equal to zero because division by zero is undefined. Therefore, we need: x0andx9x \neq 0 \quad \text{and} \quad x \neq 9

Combining these conditions:

  • From the square root condition, x8x \geq 8.
  • From the denominator, x0x \neq 0 and x9x \neq 9.

Thus, the domain of the function is x8x \geq 8, but excluding x=9x = 9.

Domain in interval notation:

[8,9)(9,)\boxed{[8, 9) \cup (9, \infty)}

Would you like further details or clarifications on any steps?

Here are some related questions to consider:

  1. What if the square root were of a different expression?
  2. How does adding other restrictions to the denominator change the domain?
  3. What happens to the domain if there is no square root involved?
  4. How does the domain change for functions involving cubes instead of square roots?
  5. How would you solve for the domain if complex numbers were allowed?

Tip: Always check for restrictions like square roots and denominators when determining the domain of a function.

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Math Problem Analysis

Mathematical Concepts

Function Domain
Square Roots
Rational Functions

Formulas

√(x - 8) for x ≥ 8
Denominator cannot be zero: x(x - 9) ≠ 0

Theorems

Conditions for the Domain of a Function

Suitable Grade Level

Grades 9-12