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A rational function is a type of function that is the ratio of two polynomials. For example, in the function you've encountered, \(h(x) = \frac{1}{x+2} \), the numerator (top part) is 1, and the denominator (bottom part) is \(x+2\).

This means that the value of the function changes based on the input value of \(x \). Rational functions can have interesting properties, like vertical asymptotes, which occur when the denominator equals zero, resulting in undefined points.

To evaluate rational functions:
- Identify the numerator and the denominator.
- Substitute the given value into the function.
- Simplify to find the result. Always be cautious of zero in the denominator, as this leads to division by zero.

Division by zero is a crucial concept in mathematics, as it is undefined. For instance, in the given function \(h(x) = \frac{1}{x+2}\), if we substitute \x = -2\, we get \(h(-2) = \frac{1}{-2+2} = \frac{1}{0}\), which is not possible.

This means we must be careful with values that make the denominator zero in rational functions. These values create vertical lines called vertical asymptotes in the graph of the function. They indicate points where the function doesn't exist. In our example, \x=-2\ is a vertical asymptote of \(h(x)\).

Remember:
- Division by zero is undefined.
- Identify values that make the denominator zero.
- Understand that these values lead to discontinuities in the function.

The substitution method involves replacing the variable in the function with a given value and then simplifying. Here’s how it works step-by-step:

1. Begin with the given function: \(h(x) = \frac{1}{x+2}\).
2. Substitute the specific value of \x\ into the function. For example, to find \h(0)\, you replace \x\ with 0:
\[h(0)=\frac{1}{0+2}=\frac{1}{2}\]
3. Simplify the expression to get the result.

Using substitution, we found that:
- \h(0) = \frac{1}{2}\.
- \h(2) = \frac{1}{4}\.
- \h(-2)\ is undefined because it leads to division by zero.

Substitution is a straightforward method to evaluate functions efficiently. Just remember to always simplify the expression after replacing the variable.