91Ó°ÊÓ

A rational function is defined as a fraction with a numerator and a denominator that are both polynomials. For our specific example, the rational function is given by:
$$f(x) = \frac{x^3 + 2}{x^2}.$$
The numerator is the polynomial: \(x^3 + 2\), and the denominator is: \(x^2\).
Rational functions can be a bit tricky because the denominator cannot be zero. If the denominator is zero, the function is undefined. Our job is to figure out which values of \(x\) make the denominator zero and exclude those from the domain. More on this in the next section.

The domain of a rational function includes all real numbers except those that make the denominator equal to zero. Essentially, we want to identify these 'trouble' values.
To find these values, we set the denominator equal to zero and solve:
\(x^2 = 0\).
Solving this equation gives us \(x = 0\). This means that \(x = 0\) is a value where the function is undefined. So, we exclude \(x = 0\) from the domain. Therefore, the domain of the function \(f(x)\) is all real numbers except \(x = 0\). You can write this in interval notation as:
\((-\infty, 0) \cup (0, +\infty)\).

In rational functions, the denominator is crucial because it determines where the function may be undefined. For our function
$$f(x) = \frac{x^3 + 2}{x^2},$$
the denominator is \(x^2\).
Whenever the denominator equals zero, the function breaks down since division by zero is not possible.
By setting \(x^2 = 0\) and solving, we find that \(x = 0\) makes the denominator zero. So, we must exclude \(x = 0\) from the domain to ensure the function remains valid.
In conclusion, always remember to check the denominator of a rational function. Identify the values that make it zero and exclude them from the domain. It’s a critical step to understand and solve problems involving rational functions effectively.