What is the domain of this function?

Question

(2x-1) / (x²+2)

I know for the domain I set the bottom to zero (or acctually NOT equal to zero) so I have x²+2 = 0 and then get x²= -2 but what then? Does the end result include i?

And also how do I find the zeros?

Answer 1

as you said, there are no real solutions to x^2=-2, so all real numbers are in the domain.

If we want to check the complex numbers as well, then we need to solve

(a+bi)^2=-2

a^2+2abi-b^2=-2

a^2-b^2=-2

2ab=0

so either a or b must equal 0.

for b=0 there is no solution to the first (real part) equation, so a=0

-b^2=-2

b=+-sqrt2

therefore the domain is

{x belongs to C|x not equal to sqrt2*i nor -sqrt2*i}

Answer 2

Hey mate,

The method you have taken is completely correct, unless otherwise stated the domain of a function is defined as the set of values for which the function is continuous, in your situation for a quoitent function p(x)/r(x), this applies to all values of x for which r(x) != 0, as you stated solving the roots of the denominator yields x^2 = -2, which in the Real Number system has no solutions, however in the complex plane yields two solutions x = sqrt(2)i, x = -sqrt(2)i, so with that said it all comes down to the nature under which your function is defined, if it is a real valued function (which it looks like), then the domain spans the entire x-axis (i.e. -oo < x < oo), as as a complex function f(z) = f(x + iy), the domain would be

X x Y | y ! = sqrt(2) & y != -sqrt(2)

Hope this helps,

David

Answer 3

This problem is all about how to express your answer.

1.  All real numbers are acceptable as input in order to get a real number answer.

2.  If it is necessary to consider complex numbers, then you are aware of which number, SQRT(-2), causes the function to go to infinity.

You should probably include two sentences to be thorough.

Answer 4

This is what's called a rational function, and they are defined for all values such that the denominator is not 0 (so you're doing the right thing so far). Now, the question is, is this a real or complex domain function? If you are just using it on the reals, then there is no number in the set of real numbers such that the denominator is 0. i.e. the domain of the real valued function is all R. If this is a complex function, then the domain will be all complex numbers except i * sqrt(2) and -i * sqrt(2), because they make the denominator 0.