If I have a function: f(x) = X^2 + 10/(X-2)
and X is 2, the second part of the equation is undefined, correct?
If I want to find the maximum domain and range of the function, how do I
Do I just ignore the undefined section because of the defined component?
Thanks for your help.
If one part of the formula is undefined at X=2 then it is likely that the function is undefined there. You should however be careful to check whether the formula can be simplified. For example, (X^2-4)/(X-2) is undefined at 2, but you can extract a factor of X-2 from numerator and denominator to get X+2, which is defined at X=2. This is a very simple example of a removable singularity.
No, you cannot do that. The function f(x) is the whole [x^2 +10/(x-2)].
You cannot separate the two terms of the [x^2 +10/(x-2)] and still call the separated terms as f(x).
When x=2, the 10/(x-2) is undefined while the x^2 is defined. That means the whole [x^2 +10/(x-2)], which is f(x), is not defined.
You want to find the maximum domain and range of f(x).
Well, the domain of f(x) is all the values of x where f(x) is defined. In the whole set of real numbers, f(x) is defined, except when x=2, so
Domain is the whole set of real numbers of x, except x=2.
The range of f(x) is the set of real numbers that what the domain can make of f(x).
We know that x=2 is not part of the domain, so f(2) is not part of the range.
What is f(2)?
f(2) = 2^2 +10/(2-2) = 4 + (undefined, or, say, infinity) = say, infinity.
Well, in any case, infinity is not part of any range of any function because infinity is not a real number.
For all the other real numbers value of x, all the real numbers value of y, including y=0, are covered.
Therefore, the range of f(x) is all real numbers.