Understanding the Function f(x) = |x – 2| + 1: Analysis, Graph, and Applications

f(x) = |x – 2| + 1

To understand and analyze the function f(x) = |x – 2| + 1, let’s break it down step by step

To understand and analyze the function f(x) = |x – 2| + 1, let’s break it down step by step.

The function |x| represents the absolute value of x, which is defined as the distance of x from the origin on the number line.

In our case, f(x) = |x – 2| + 1, means that we take the absolute value of (x – 2), which represents the distance between x and 2, and then add 1 to it.

To analyze this function, let’s consider a few scenarios:

1. When x < 2: In this case, (x - 2) becomes negative, resulting in a negative distance. However, taking the absolute value will make it positive. So, we have |negative value| = positive value. Therefore, f(x) = |x - 2| + 1 will give us a positive value greater than 1. 2. When x = 2: At x = 2, (x - 2) becomes equal to zero, which means the distance between x and 2 is zero. Thus, the absolute value of zero is also zero. Adding 1 to it gives f(x) = 1. 3. When x > 2:
For x values greater than 2, (x – 2) becomes positive, representing a positive distance. The absolute value will keep it positive. Consequently, f(x) = |x – 2| + 1 will provide a positive value greater than 1.

In summary, the function f(x) = |x – 2| + 1 has a graph that consists of a “V” shape, centered at x = 2, with the vertex at (2, 1). The function will always output values greater than or equal to 1, as the absolute value ensures non-negative results.

By understanding the behavior of this function, you can now solve equations and inequalities involving f(x), graph the function, or analyze its properties.

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