The domain of a function refers to all the possible values of the independent variable, usually represented by \(x\), for which the function is defined. In simpler terms, it tells us what \(x\) values can be plugged into a function without causing any mathematical mishaps, like dividing by zero.
For the function \(f(x) = \frac{3x^2 + x - 5}{x^2 + 1}\), determining the domain involves checking the denominator, \(x^2 + 1\). A crucial detail in rational functions is that the denominator must not be zero because division by zero is undefined. Here, when solving \(x^2 + 1 = 0\), we realize \(x^2 = -1\) has no real solution, as squares of real numbers are never negative.
This means \(x\) can indeed be any real number because there are no restrictions from the denominator. Therefore, the domain of this function is all real numbers, denoted by \( (-\infty, \infty) \).
- Ensure the denominator never equals zero.
- The domain is the set of all permissible \(x\) values.