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The domain of a function represents all the possible input values, typically "x-values," for which the function is defined. For the function \( f(x) = \frac{x}{x^2 - 2x - 8} \), we need to ensure that the denominator is not zero to avoid undefined expressions.

To find this, set the denominator equal to zero and solve: \[ x^2 - 2x - 8 = 0 \] Factoring, we get: \((x-4)(x+2) = 0\). Thus, the values that make the denominator zero are \(x = 4\) and \(x = -2\). These are the points at which the function is undefined.

Thus, the domain of \( f(x) \) is all real numbers except \( x = 4 \) and \( x = -2 \). Use interval notation to express this as:

• \( (-\infty, -2) \)
• \((-2, 4)\)
• \((4, \infty)\)

The First Derivative Test is a method used to determine the relative extrema (maxima or minima) of a function using its derivative. By examining where the derivative changes its sign, you can identify points where the function changes from increasing to decreasing or vice versa.

For the function \( f(x) = \frac{x}{x^2 - 2x - 8} \), we start by finding the derivative \( f'(x) \) using the quotient rule. Simplifying, we find: \[ f'(x) = \frac{-x^2 - 8}{(x^2 - 2x - 8)^2} \] To identify critical points, we find where \( f'(x) = 0 \), but there are no real solutions. Instead, check where the derivative is undefined, which occurs at \( x = 4 \) and \( x = -2 \).

Using these findings to apply the First Derivative Test, determine if changes in sign occur around these points:

• At \( x = -2 \): If \( f'(x) \) goes from negative to positive, there is a relative minimum.
• At \( x = 4 \): If \( f'(x) \) goes from positive to negative, there is a relative maximum.

This analysis helps identify behavior changes in the function at critical points.

Understanding the intervals on which a function is increasing or decreasing is essential for graphing and analyzing functions. This involves examining the sign of the first derivative \( f'(x) \). Intervals where \( f'(x) > 0 \) indicate that the function is increasing, while \( f'(x) < 0 \) indicates it is decreasing.

Using test points, examine \( f'(x) = \frac{-x^2 - 8}{(x^2 - 2x - 8)^2} \) in each interval:

• For the interval \((-\infty, -2)\), choose a test point like \(x = -3\). The derivative is negative, indicating the function is decreasing.
• In \((-2, 4)\), use \(x = 0\) as a test point. The derivative is positive, showing the function is increasing.
• In the interval \((4, \infty)\), use a test point like \(x = 5\). Here, the derivative is negative, demonstrating the function is decreasing.

Knowing these intervals provides insight into the overall behavior of the function in different regions, allowing us to determine the overall shape of \( f(x) \) when graphed.