Implied domain

[Snorlax]

Hey guys, stuck on a question.

f(x) = √x^2 - 7x +12
      = √(x-3)(x-4)

Now graphing it, I dont understand why the x=3 has the domain of (-∞, 3].
Why doesn't it go [3,∞)?

Also, how do you write mathematically on AN?
Like, the bold format I see people helping others with.

Thanks.

[brightsky]

Your function is of the form f(x) = sqrt(blah). You know that blah cannot be negative. So to find domain, we let blah be greater or equal to 0.

(x-3)(x-4) >= 0 [That is a greater or equal to sign by the way.]

Sketching the parabola, we notice that y is greater or equal to 0 when x is less than or equal to 3 and when x is greater than or equal to 4. Therefore, the domain is (-infinity, 3] U [4, + infinity).

Now your proposal is [3, + infinity). A good way to check whether this is correct is to substitute in some random x value in that interval and see if the function exists. Now we know that 3.5 is in that interval. We substitute that in:

f(3.5) = sqrt((3.5-3)(3.5-4)) = sqrt(-0.25). We have a negative number under the square root. Bad news. The function does not exist.

You write mathematically on AN using latex. There should be a guide somewhere on the forum...

[Snorlax]

oh thanks
So because x cannot be >3 or <4, you imply their domains.
right.
How stupid of me. Even the topic is IMPLIED domains...

[brightsky]

Haha yeah implied domain means the maximum possible domain for a given function. So you just work out ALL the x-values for which the function is defined and there's your implied domain.