Domain and Range Clarification

[sensitiverabbit]

For this question, it says to "Find the domain of the following functions using interval notation". The answer is boxed, but I was wondering why the answer wasn't "x ≠ 2" since when you plug in 2 it becomes "f(x)=0" which is incorrect. Please clarify for me, thank you!

 

[topsquark]

For this question, it says to "Find the domain of the following functions using interval notation".View attachment 33427 The answer is boxed, but I was wondering why the answer wasn't "x ≠ 2" since when you plug in 2 it becomes "f(x)=0" which is incorrect. Please clarify for me, thank you!

To be clear,[imath]x \neq 2[/imath] in interval notation is [imath]( -\infty, 2) \cup (2, \infty )[/imath].

But f(2) certainly exists, so why would you exclude it? A number x does not belong to the domain only if the function value does not exist at it. So if we had [imath]f(x) = \dfrac{1}{x^2 - 4x + 4}[/imath] then we can't have x = 2 because we can't calculate f(2).

-Dan

 

[Otis]

wondering why the answer wasn't "x ≠ 2" since when you plug in 2 it becomes "f(x)=0" which is incorrect.

Hi sensitiverabbit. Function f is defined as a polynomial. The domain for all polynomial functions is the set of Real numbers.

When you say that f(2)=0 is incorrect for the domain, it makes me think that you're working with a bad definition for "domain".

A function's domain is the set of all x-values for which the function is defined. The function you posted IS defined at x=2, so 2 must be in the domain. In fact, we may substitute any Real number for x, and the polynomial will evaluate to a Real number. There is no value for which f is not defined.



[imath]\;[/imath]

 

[Steven G]

x ≠ 2" since when you plug in 2 it becomes "f(x)=0" which is incorrect

I am assuming that you meant to say that f(2) =0 which is incorrect.
So you think f(2)≠0, so what is incorrect?
For the record, f(2) = 0 and 0 is a real number!. As long as the x-value you plug into the function gives you back a real number, then that x-value is in the domain.