what is the domain of 1-x^2/x and what are the steps

Question

I need to know the domain and the steps involved in finding the domain for the expression 1-x^2/x

Answer 1

The domain is basically all numbers you can plug into the function that doesn't make the function undefined (in other words it doesn't break the function). 

 

For a function with no defined boundaries in the domain (ie x>=0), you generally start out with x=all real numbers (-inf < x < inf) and then try to find points or range of numbers where plugging into the function doesn't make sense. Classic examples are

 

1) you can never have a denominator equal to zero

2) you can never take a log or ln of zero

 

Any number that gives you those 2 scenarios are automatically outside the domain of the function. 

 

In the case of your function 1-x^2/x. You see there's a fraction in the function. Therefore, the function will breakdown (become undefined) if the denominator equals zero. That happens when x=0. Therefore x=0 is not in the domain of the function. Otherwise, any other number when plugged in to the function will yield an answer. Thus the domain is x=all real numbers except 0. Or you can just specify that the domain is x not equal to 0.

 

Besides the general scenarios like the ones mentioned above, you may also be asked about domains of composition of two or more functions (ie g(f(x))). In that case, the domain of g(f(x)) is restricted by the domain of f(x) meaning a number cannot be in the domain of g(f(x)) if it is not in the domain of f(x) even if that number is in the domain of g(x). 

 

Let me use an example to illustrate:

 

Suppose g(x) = 1/(x+1) and f(x) = 2/x

 

Then g(f(x)) = 1/[(2/x)+1] = 1/[(x+2)/x] = x/(x+2)

 

Even though x=0 is in the domains of both 1/(x+1) and x/(x+2) and g(f(x)) = x/(x+2), x=0 is not in the domain of g(f(x)), because x=0 is not in the domain of f(x). 

 

 

Answer 2

I can't tell if you mean (1 - x²) / x or 1 - (x²/x). But in fact it doesn't matter. 

 

Remember that the domain is the set of numbers for which the function is defined.

 

To find the domain of a rational function, you start with the set of all real numbers, and then determine any real numbers for which the function is NOT defined.  Generally that means any numbers for which the denominator is 0.

 

In this case, there is the same denominator no matter which of the two possibilities is correct.  That is, x.  If x = 0, then the function is not defined.  But it IS defined for all other real numbers.

 

So the answer is, all real numbers except 0.