# Domain of a Function

#### joshuaa

How to find the domain of the function.

3/(1 +(4/x^2))^(1/2)

when I set the denominator equal to zero, I get stuck

1 + (4/x^2) = 0
x^2 + 4 = 0
x^2 = -4
now cannot take the root of negative

#### romsek

MHF Helper
what happens when you set $x=0$ ?

#### joshuaa

the denominator becomes (1 + 4/0)^(1/2)

which is undefined

#### romsek

MHF Helper
the denominator becomes 1 + 4/0

which is undefined
exactly, so $0$ cannot be in the domain.

Do you see any other numbers which cannot be in the domain?

#### joshuaa

negative values will be positive, so I don't see any other numbers

but is it possible to multiply numerator and denominator by x like this?

3x/x(1 +(4/x^2))^(1/2)

then I enter the x inside the root x^2 which will lead to

3x/(x^2 + 4)^(1/2)

then I will have no problem when the x = 0

#### romsek

MHF Helper
negative values will be positive, so I don't see any other numbers

but is it possible to multiply numerator and denominator by x like this?

3x/x(1 +(4/x^2))^(1/2)

then I enter the x inside the root x^2 which will lead to

3x/(x^2 + 4)^(1/2)

then I will have no problem when the x = 0
if $x=0$ can you multiply by $\dfrac 1 x$

is $\dfrac 1 x$ even defined?

#### joshuaa

I could not and it was not defined before, but now I did some algebra which helped me get rid of 1/x^2

#### romsek

MHF Helper
I could not and it was not defined before, but now I did some algebra which helped me get rid of 1/x^2
yeah but that algebra you did involved dividing $x$ which is undefined when $x=0$

when x = 0

now

0/(4)^(1/2) = 0

#### romsek

MHF Helper
when x = 0

now

0/(4)^(1/2) = 0
you start off with

$\dfrac{3}{\sqrt{1+\frac{4}{x^2}}}=\dfrac{3}{\sqrt{ \dfrac{4+x^2}{x^2}}}=\dfrac{3}{\dfrac{\sqrt{4+x^2}}{|x|}}$

now you cleverly multiply by 1 in the form of $\dfrac {|x|}{|x|}$ to obtain $\dfrac{3|x|}{\sqrt{4+x^2}}$

but when $x=0$ the expression $\dfrac {|x|}{|x|}$ is undefined and thus your bit of cleverness is not allowed.