# How do you find the domain of #f(x) = ((x + 8)^(1/2)) /( (x + 3)(x - 2))#?

You can rewrite the given expression as:

Then, since you cannot calculate the square root of a negative number and you cannot divide by zero, the domain is the solution of the following conditions:

that's

that can be written as:

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To find the domain of ( f(x) = \frac{\sqrt{x + 8}}{(x + 3)(x - 2)} ), we need to identify any values of ( x ) that would make the function undefined.

The function will be undefined if:

- The denominator is equal to zero.
- The argument of the square root function is negative because square roots of negative numbers are not real.

First, we address the denominator: [ (x + 3)(x - 2) = 0 ] This gives us the values of ( x ) where the denominator would be zero: ( x = -3 ) and ( x = 2 ).

Now, for the argument of the square root: [ x + 8 \geq 0 ] which implies: [ x \geq -8 ]

Therefore, the domain of the function is all real numbers except ( x = -3 ) and ( x = 2 ), because they would make the denominator zero, and ( x < -8 ) because it would make the argument of the square root negative.

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