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

Answer 1

#x in [-8;-3)uu(-3;2) uu (2;+oo)#

You can rewrite the given expression as:

#sqrt(x+8)/((x+3)(x-2)#

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:

#x+8>=0 and x+3!=0 and x-2!=0#

that's

#x>=-8 and x!=-3 and x!=2#

that can be written as:

#x in [-8;-3)uu(-3;2) uu (2;+oo)#
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Answer 2

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:

  1. The denominator is equal to zero.
  2. 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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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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