91Ó°ÊÓ

Rational functions are expressions formed by dividing two polynomials, where the polynomial in the denominator cannot be zero. These functions can exhibit interesting behavior based on the values that make the denominator zero. For instance, functions of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, have undefined points when \( Q(x) = 0 \). In simpler terms, these points are where the value of the function shoots towards infinity or is undetermined.
In the case of our rational function \( h(w) = \frac{w+7}{w^2-81} \), the polynomial \( w^2-81 \) as the denominator should not be zero, thereby determining the domain where the function behaves well and is defined. By understanding this, we can easily pinpoint intervals where the function is legitimate.

When dealing with polynomials, one important factorization technique is the difference of squares. This technique is applied when you have an expression of the form \( a^2 - b^2 \). It can be factored as \( (a-b)(a+b) \). This is crucial, as it helps simplify expressions or solve equations.
Consider the polynomial \( w^2-81 \) in our example. Recognizing it as a difference of squares, it can be factored to \( (w - 9)(w + 9) \). Factoring makes it straightforward to find the roots which are \( w = 9 \) and \( w = -9 \). These roots are pivotal as they are points where the original function \( h(w) \) becomes undefined.

Interval notation is a mathematical technique to describe ranges of numbers or intervals on the real number line succinctly. It is especially useful in describing domains of functions. In interval notation, parentheses \(()\) are used to denote that endpoints are not included, also known as open intervals, whereas brackets \([]\) indicate included endpoints or closed intervals.
For the function \( h(w) = \frac{w+7}{w^2-81} \), the denominator becoming zero at \( w = 9 \) and \( w = -9 \) make these points excluded from the domain. Expressing its domain in interval notation, we can say:
\[ (-\infty, -9) \cup (-9, 9) \cup (9, \infty) \]
This shows clearly that all values of \( w \) except \( -9 \) and \( 9 \) are valid inputs for this function.

Undefined expressions occur in mathematics when operations such as division by zero are attempted. In rational functions, any value that makes the denominator zero results in an undefined expression. These undefined points make the function discontinuous or non-existent at those spots.
Using our example \( h(w) = \frac{w+7}{w^2-81} \), when \( w \) is either \( 9 \) or \( -9 \), both make the denominator zero, confirming that \( h(w) \) is undefined at these points. Awareness of undefined expressions is crucial for understanding the complete behavior and limitations of rational functions.