91Ó°ÊÓ

Determining the domain of a function is an important step in understanding its behavior. The domain of a function includes all the possible input values (x-values) for which the function is defined. For rational functions, such as \( f(x) = \frac{3}{x^2 - 4} \), the denominator cannot be zero as it would make the function undefined.
To find when \( f(x) \) is undefined, set the denominator equal to zero and solve for \( x \):
\[ x^2 - 4 = 0 \]
Solving this gives \( x^2 = 4 \), therefore \( x = \pm 2 \). This means the function is undefined when \( x = -2 \) and \( x = 2 \).

So, the domain of \( f(x) \) is all real numbers except \( x = -2 \) and \( x = 2 \), which we can express as:
\[ \text{Domain of } f(x): \{ x \in \mathbb{R} \mid x eq -2, 2 \} \]
Understanding these domain restrictions helps to avoid calculation errors and gives a clearer picture of how the function behaves across different values.

An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. This difference is known as the common difference. In the context of our exercise, the sequence of \( x \) values follows an arithmetic sequence.

The given sequence starts at \( -3 \) and increases by \( 1 \) (the common difference) each time, forming the sequence \( -3, -2, -1, 0, 1, 2, 3 \). This arithmetic progression organizes the way we evaluate the rational function \( f(x) \) across a range of \( x \) values.

Identifying the arithmetic sequence in tasks like this assists in calculating values efficiently and ensuring all relevant x-values are considered.

• The first term is \( -3 \)
• The common difference is \( +1 \)
• Subsequent terms are created by adding \( 1 \) to the previous term

The understanding of arithmetic sequences is a useful tool in solving mathematical problems systematically.

A table of values is a simple but vital tool to visualize how a function behaves over a set of inputs, helping you observe patterns, calculate function outputs, and identify undefined points.

For the function \( f(x) = \frac{3}{x^2 - 4} \), we use a table to calculate the output \( f(x) \) for each input \( x \) value in the series \(-3\) to \(3\).

Here's how you fill the table:

• Start with the first \( x \): if \( x = -3 \), calculate \( f(-3) = \frac{3}{5} \)
• If a calculated \( x \) (like \( -2 \)) makes the denominator zero, note that \( f(x) \) is undefined there
• Continue this for each \( x \) value until \( 3 \)

Tables of values allow you to systematically handle variable replacements and maintain results in an organized way. They serve as a handy reference to spot trends or confirm calculations in exercises and problems.