What is the domain of this function?

In summary, the domain of the function \sqrt{{x-3}\over{x-1}} can be determined in two ways. The first way considers the expression {{x-3} \over {x-1}} \geq 0 and yields a domain of \left(-\infty, 1\right) \cup \left[3, \infty\right). The second way interprets the function as \sqrt{{x-3}\over{x-1}} = \frac{\sqrt{x-3}}{\sqrt{x-1}} and takes into account the conditions x \geq 3 and x > 1, resulting in a domain of \left[3, \infty\right).
  • #1
murshid_islam
457
19
suppose x is a real variable. what is the domain of the function [tex]\sqrt{{x-3}\over{x-1}}[/tex]?

here [tex]{{x-3} \over {x-1}} \geq 0[/tex]
from this we get the domain to be [tex]\left(-\infty, 1\right) \cup \left[3, \infty\right)[/tex]

but if we write the function [tex]\sqrt{{x-3}\over{x-1}} = \frac{\sqrt{x-3}}{\sqrt{x-1}}[/tex]

then [tex]x-3 \geq 0 \Rightarrow x \geq 3[/tex]
and [tex]x-1 > 0 \Rightarrow x > 1[/tex]

from these two, we get the domain [tex]\left[3, \infty\right)[/tex]

why do i get two different domains?
 
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  • #2
Because in the first instance you let yourself evaluate a/b first and if the answer is positive we can take the square root. In the second you want to take the square roots of a and b first. Obviously a/b is not only positive when a and b are.
 
  • #3
Furthermore, the maximal domain for the last function starts at 3, not 1.
 
  • #4
arildno said:
Furthermore, the maximal domain for the last function starts at 3, not 1.
you are right. i have edited my previous post now.
 
  • #5
but how can the same function has different domains? can anybody please explain?
 
  • #6
murshid_islam said:
but how can the same function has different domains? can anybody please explain?
Because it isn't the same function! :smile:
 
  • #7
murshid_islam said:
but how can the same function has different domains? can anybody please explain?
The idea that 'the domain of a function' is what you can work out by 'making sure we don't divide by zero or square root a negative number' at any point is completely anad utterly wrong. It is not you who is at fault but a moronic misapprehension of the definition of function that is propogated throughout the education system of many parts of the supposedly educated world.

The domain of a function is part of its definition. What you wrote out are two different expressions which when interpreted in some manner define functions that have different domains according to some particularly idiotic ideas of what a domain is, and that, when evalauate on the intersection of the 'domians' agree.
 
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  • #8
Actually, this is yet another case of something that a lot of your posts embody: often you perform some manipulation and get an answer that contradicts your intuition. You then ask us why the answer is what it is when really you should not be asking us anything but asking yourself 'is my intuition at all correct'.
 
  • #9
matt grime said:
Actually, this is yet another case of something that a lot of your posts embody: often you perform some manipulation and get an answer that contradicts your intuition. You then ask us why the answer is what it is when really you should not be asking us anything but asking yourself 'is my intuition at all correct'.
well, i know that i am a moron. but i want to learn. that's why i ask stupid questions. i have learned a lot from this forum by asking stupid questions. not everyone is as skilled in maths as you are. so try to be a little patient with the unfortunate (or dumb) ones. :smile:
 
  • #10
well...if you take the square root of a ngative over a negative you'd end up with a bunch of i which would cancel making the function real again.

[tex]\sqrt{\frac{a} {b}} = \frac{\sqrt{a}} {\sqrt{b}}[/tex] regardless of a and b as long as a/b >0 or a=0
 
  • #11
murshid_islam said:
well, i know that i am a moron. but i want to learn. that's why i ask stupid questions. i have learned a lot from this forum by asking stupid questions. not everyone is as skilled in maths as you are. so try to be a little patient with the unfortunate (or dumb) ones. :smile:

I didn't call you a moron. I didn't insult you at anypoint. I said two things.

1. The concept of domain is mistaught a hell of a lot
2. Ask yourself 'why did I expect something different each time'
 
  • #12
matt grime said:
I didn't call you a moron. I didn't insult you at anypoint.
i never said that you insulted me in any way. i am sorry for any misunderstandings.
 

What is the domain of a function?

The domain of a function is the set of all possible input values that can be plugged into the function to produce an output. It is often represented by the variable "x" and can include all real numbers or a restricted set of values depending on the function.

Why is it important to know the domain of a function?

The domain of a function is important because it tells us which values we can use as input and which values we cannot. This helps us avoid errors and ensures that we are working within the appropriate range for the function.

How do you determine the domain of a function?

To determine the domain of a function, you must look for any restrictions on the input values. This could include square roots, division by zero, or negative numbers under odd roots. You must also consider the context of the function, such as a real-world scenario, to determine any practical limitations on the domain.

Can the domain of a function be negative?

Yes, the domain of a function can include negative values. It ultimately depends on the specific function and its restrictions. Some functions may have a limited or restricted domain that only includes positive values, while others may have a wider range that includes negative values.

What happens if you choose an input value outside of the domain?

If you choose an input value outside of the domain, the function will not be defined for that value. This means that there will be no corresponding output and the function will be undefined. It is important to stay within the domain in order to accurately use and interpret the function.

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