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 \( f(x) = |x - 2| + 1 \) is composed of 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. Next, consider the behavior of the function on either side of the vertex. For \( x < 2 \), the function \( f(x) = -(x - 2) + 1 = -x + 3 \) is a linear function with a negative slope, indicating that the function is decreasing in this interval.
3. For \( x > 2 \), the function \( f(x) = (x - 2) + 1 = x - 1 \) is a linear function with a positive slope, indicating that the function is increasing in this interval.
4. Identify the largest interval where the function is increasing. Since the function is increasing for \( x > 2 \), the largest interval on which \( f(x) \) is increasing is \( (2, \infty) \).
5. Finally, confirm that the function is not increasing at \( x = 2 \) itself, as this is the vertex of the V-shape where the slope changes from negative to positive. Therefore, the interval \( (2, \infty) \) is indeed the largest interval where the function is increasing.