To determine whether the function $f(x) = \frac{x^2 - 4}{x^2 + 6}$ is odd, even, or neither, we can check the following:

1. Even Function Check:

A function $f(x)$ is even if: $f(-x) = f(x) \text{ for all } x$

2. Odd Function Check:

A function $f(x)$ is odd if: $f(-x) = -f(x) \text{ for all } x$

Evaluate $f(-x)$:

Let's substitute $-x$ into the function: $f(-x) = \frac{(-x)^2 - 4}{(-x)^2 + 6}$ Since $(-x)^2 = x^2$, we have: $f(-x) = \frac{x^2 - 4}{x^2 + 6}$ This is exactly the same as $f(x)$. Therefore: $f(-x) = f(x)$

Conclusion:

The function $f(x) = \frac{x^2 - 4}{x^2 + 6}$ is even.

Implications for Graphing:

Since the function is even, it is symmetric with respect to the y-axis. This means that the left side of the graph (for negative $x$) will be a mirror image of the right side (for positive $x$). Knowing this symmetry can significantly simplify the sketching process because you only need to analyze and plot the function for non-negative values of $x$, and then reflect the graph across the y-axis.

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Do you have any more questions or need further clarification on how this affects the graph?

Here are five questions that can deepen your understanding:

1. How does knowing that a function is even affect the way you analyze the graph?
2. Can a function be both odd and even? Why or why not?
3. What kind of symmetry would you expect if the function were odd?
4. How would you use symmetry to check for potential errors in your graph?
5. How does the symmetry in even functions relate to their domain and range?

Tip: Always check for symmetry when sketching graphs. It can often halve the amount of work you need to do, especially for even functions!