Pre-Calculus

Determine the open intervals on which the function is increasing, decreasing, or constant. f(x) = sqrt(x^2 - 9)

Step-by-step solution with explanation

Final Answer

Decreasing on ; Increasing on ; Never constant. (Domain: or )

Step-by-step solution

1

State the domain of the function

, domain: or
The expression under a square root must be non-negative. Solving gives or . The function only exists on these intervals.
2

Find the derivative using the chain rule

We differentiate using the chain rule. The outer function is the square root and the inner function is . This derivative tells us the slope of at any point in the domain.
3

Find critical points inside the domain

; denominator always on domain, so sign depends on
The denominator is always positive on the open domain intervals. So the sign of depends entirely on the sign of (the numerator). There are no points where inside the open intervals or .
4

Analyze sign of derivative on each interval

For , the numerator is negative, making negative — the function is decreasing. For , the numerator is positive, making positive — the function is increasing.
5

State the increasing and decreasing intervals

We use open intervals because at the derivative is undefined (denominator is zero). There are no intervals where the function is constant, since is never zero on the domain.

Understanding this problem

Learning Insight

The sign of the derivative tells us the behavior of the function: negative means the graph falls left to right (decreasing), positive means it rises (increasing). This function is a hyperbola-shaped curve symmetric about the y-axis, so its two branches naturally go in opposite directions.

Quick Tip

After finding , just check the sign of the numerator on each domain interval — if the denominator is always positive, the numerator controls everything.

Common Mistake

Students often include the endpoints in the intervals, but since is undefined there (division by zero), the intervals must be open: and .