Finding the domain of an equation

[Centurion1]

Homework Statement

Find the Domain of f

f(x)=2+7x (2+7x is in square root)

Homework Equations

The Attempt at a Solution

I'm really not sure. The possible answers i think of seem far to easy. This problem's annoying me because i remember doing it in alg. 2 but it was so long ago i cannot remember exactly how.

Anyway here is my attempt
f(x)= 2+7X (square root all)
now do you just plug in numbers. And if numbers match up you can say whether it is all real numbers or if there is a limit?

 

[Cyosis]

You can see the domain as all possible input values of a function. Putting a value of the domain into the function will yield a value in its range. When they ask you what the domain is of f without any other information they mean the natural domain of f which is the largest domain.

For example we have the function f(x)=1/x. If we take x=0 we get f(0)=1/0 which is not defined. Therefore x=0 is not in the domain of f. For all other values of x the function returns a defined value, therefore the domain of f are all values of x except 0. Try to apply this to your function.

Anyway here is my attempt
f(x)= 2+7X (square root all)
now do you just plug in numbers. And if numbers match up you can say whether it is all real numbers or if there is a limit?

You want to find a constraint on the x values yes, but you can do that more efficiently than just plugging in an infinite amount of numbers. You know that you cannot take the square root of a negative number this puts a constraint on the argument of the root. Can you write this down mathematically?

 

[Centurion1]

so at what point do i stop entering in new values. I have done 0-5 and they all are all defined. So is the answer to this problem all real numbers?

 

[Cyosis]

Since I just edited my first post you may not have read the part I edited in. If you haven't do read it.

No the domain does not consist of all real numbers. Try x=-1 and tell me what f(-1) is.

 

[Centurion1]

Yeah, just read your edited in part. Seeing as i am doing calculus this year i hope i can...

so would it be x>-1

 

[Cyosis]

No you're just guessing now I could ask you to try x=-0.99999 and you would run into the same problem. We don't want to just guess we want to find a systematic way of solving this problem.
Try to answer the following question.

The argument of the root has to be ...? (for clarification with the argument of the root I mean the stuff that is underneath the root sign)
If you can't answer this question then explain to me why x=-1 does not belong to the domain.

 

[Centurion1]

Well it has to be greater than any negative number if you want a defined answer. would it be greater than or EQUAL to 0? Or is that answer format just wrong?

 

[Cyosis]

Indeed the argument has to be greater than or equal to 0. We can write this down mathematically as $2+7x \ge 0$. Can you find all values for x now?

 

[Centurion1]

well wouldn't it be 0 to infinity? I could input a million or a trillion and i would get a defined answer right.

 

[Cyosis]

The interval 0 to infinity is certainly part of its domain, but it's not complete. There are more values that lie within its domain. I will say it again we know that $2+7x \ge 0$. Now how do you usually find values for x when you're given such an equality/inequality?

 

[Centurion1]

would you try to solve it like an equation? basically, (sqr. root)2+7x=0

You get -2/7. which equals zero when plugged in.

 

[Cyosis]

Basically, however you said correctly in a previous post that only the argument needs to be greater than or equal to 0. So don't solve for the square root of the argument, although the answer doesn't change. By changing the inequality into an equality you've found the minimum value of x. Do you understand this? What is the function's domain?

 

[Centurion1]

-2/7<(or equal to)x<(or equal to) infinity?

 

[Cyosis]

Almost, x can never become infinity so you need to change your smaller or equal than sign to ..?

 

[Centurion1]

so it would be less than infinity.

 

[Cyosis]

Yep.

 

[Centurion1]

thank you very much for your help (and patience).

 

[Cyosis]

You're welcome.

 

[Mark44]

-2/7<(or equal to)x<(or equal to) infinity?

In other words, the domain of your function is the set {x | -2/7 <= x < $\infty$}