Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = (x + 3)²

Verified step by step guidance

1. Identify the function given: \(f(x) = (x + 3)^2\). Our goal is to find where this function is increasing and where it is decreasing.
2. Find the first derivative of the function, \(f'(x)\), because the sign of the derivative tells us where the function is increasing or decreasing. Use the power rule: \(f'(x) = 2(x + 3)\).
3. Set the derivative equal to zero to find critical points: \$2(x + 3) = 0\(. Solve for \)x$ to find the critical point(s).
4. Use the critical point to divide the number line into intervals. Test a value from each interval in the derivative \(f'(x)\) to determine if the function is increasing (where \(f'(x) > 0\)) or decreasing (where \(f'(x) < 0\)) on that interval.
5. Summarize the intervals where \(f(x)\) is increasing and where it is decreasing based on the sign of \(f'(x)\) in each interval.

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions like ƒ(x) = (x + 3)^2, the domain is all real numbers since there are no restrictions such as division by zero or square roots of negative numbers.

Increasing and Decreasing Intervals

A function is increasing on an interval if its output values rise as the input values increase, and decreasing if its output values fall as the input values increase. Identifying these intervals helps understand the behavior of the function over different parts of its domain.

Using the Derivative to Determine Monotonicity

The derivative of a function indicates the slope of the tangent line at any point. If the derivative is positive over an interval, the function is increasing there; if negative, the function is decreasing. For ƒ(x) = (x + 3)^2, finding and analyzing ƒ'(x) helps determine where the function increases or decreases.