f(x)=(3x-6)/(x^2-25)
Hi Melyssa!
In order to identify the domain of a function (the possible x values that we can plug into the function), we need to figure out if there are any x values that would violate what I call the "laws of math" dealing with real numbers:
Law #1: You cannot divide by 0
Law #2: You cannot take the square root of a negative number
With that in mind, the domain of any function is all real numbers, EXCEPT any x values would violate those laws.
Are there any values of x that would violate those laws?
There are no square roots in the equation, so we don't have to worry about Law #2.
What about Law #1? We can't divide by 0, so the denominator of the function cannot be equal to 0. Let's figure out what x values would make the denominator equal to 0, and we will know that those x values will NOT be part of the domain (because they would violate the Law about dividing by 0)
x^{2} - 25 = 0
x^{2} = 25
√(x^{2}) = √(25)
x = +/- 5 (remember, the square root of a number can be either positive or negative...)
So, the domain of the function would be all real numbers, except x≠-5 or 5