Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = x² - 4x + 3

Verified step by step guidance

1. Identify the function given: \(f(x) = x^2 - 4x + 3\). To analyze where the function is increasing or decreasing, we first need to find its derivative.
2. Find the derivative of the function using the power rule: \(f'(x) = 2x - 4\).
3. Set the derivative equal to zero to find critical points: \$2x - 4 = 0\(. Solve for \)x$ to find the critical value(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 \(f'(x) > 0\) as the intervals where \(f(x)\) is increasing, and where \(f'(x) < 0\) as the intervals where \(f(x)\) is decreasing.

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) = x² - 4x + 3, the domain is all real numbers since polynomials are defined everywhere on the real line.

Increasing and Decreasing Functions

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 increases. Identifying these intervals helps understand the function's behavior and graph shape.

Using the First Derivative to Determine Intervals

The first derivative of a function indicates the slope of the tangent line. If the derivative is positive on an interval, the function is increasing there; if negative, the function is decreasing. Finding critical points where the derivative is zero helps partition the domain into these intervals.