Domain of Rational Functions

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mathdad

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1. Find domain of y = 1/(x^2 + 4).

Setting the denominator to 0 yields x^2 = - 4.
The solution(s) is found in complex roots.

The answer online given is:

Here we have no solution, so the denominator is never zero. Then the domain is "all x".

I do not understand why the domain is all x.

2. Find domain of R(x) = x^3/(x^2 + 1).

The answer in the textbook is:

The domain of R(x) is the set of all real numbers. Why? I thought the domain is found in complex roots.
 
... we have no [Real solution for x^2=-4], so the denominator is never zero. Then the domain is "all x".

I do not understand why the domain is all x ...
It's because no value of x leads to division by zero, so the rational function is defined for all Real numbers. The variable x is measured on the x-axis (i.e., the Real number line). No point on the x-axis represents an imaginary number, so the denominator is never zero.

?
 
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