Find the largest interval on which the given function is increasing.


a. ƒ(x) = |x - 2| + 1

Verified step by step guidance

1. First, understand that the function ƒ(x) = |x - 2| + 1 is an absolute value function, which creates a V-shape graph. The vertex of this V-shape is at the point where the expression inside the absolute value is zero, i.e., x = 2.
2. To determine where the function is increasing, we need to analyze the behavior of the function on either side of the vertex. For x < 2, the function behaves as ƒ(x) = -(x - 2) + 1, and for x > 2, it behaves as ƒ(x) = (x - 2) + 1.
3. Calculate the derivative of the function in each piece to determine the slope. For x < 2, the derivative is ƒ'(x) = -1, indicating a decreasing function. For x > 2, the derivative is ƒ'(x) = 1, indicating an increasing function.
4. The function is increasing where the derivative is positive. From the derivative analysis, we see that the function is increasing for x > 2.
5. Thus, the largest interval on which the function is increasing is (2, ∞).

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of x from zero on the number line, always yielding a non-negative result. For the function ƒ(x) = |x - 2| + 1, the expression |x - 2| indicates that the function's behavior changes at x = 2, where it transitions from decreasing to increasing or vice versa.

Critical Points

Critical points are values of x where the derivative of a function is zero or undefined. These points are essential for determining intervals of increase or decrease. For the function ƒ(x) = |x - 2| + 1, identifying critical points involves analyzing where the derivative changes sign, which helps in finding where the function is increasing.

Increasing and Decreasing Intervals

An interval is considered increasing if the function's output rises as the input increases. To find these intervals, one must evaluate the sign of the derivative across the critical points. For ƒ(x) = |x - 2| + 1, determining where the derivative is positive will reveal the largest interval on which the function is increasing.