1. (extra credit 10 pts.) Consider the function of two variables given by \( f(x, y)=\frac{1}{2}(x+y+|x-y|) \) Experiment with \( f \) on scratch paper for various real-number inputs in plac \( x \) and \( y \). Show two of the calculations below and briefly explain what this function does.

To analyze the function \( f(x, y) = \frac{1}{2}(x + y + |x - y|) \), we will evaluate it for two different pairs of real-number inputs. ### Step 1: Understanding the Function The function consists of three parts: 1. \( x \) 2. \( y \) 3. \( |x - y| \) (the absolute difference between \( x \) and \( y \)) The absolute value \( |x - y| \) will affect the output based on the relationship between \( x \) and \( y \). ### Step 2: Evaluating the Function #### Example 1: \( x = 3, y = 1 \) 1. Calculate \( |x - y| \): \[ |3 - 1| = |2| = 2 \] 2. Substitute into the function: \[ f(3, 1) = \frac{1}{2}(3 + 1 + 2) = \frac{1}{2}(6) = 3 \] #### Example 2: \( x = 1, y = 4 \) 1. Calculate \( |x - y| \): \[ |1 - 4| = |-3| = 3 \] 2. Substitute into the function: \[ f(1, 4) = \frac{1}{2}(1 + 4 + 3) = \frac{1}{2}(8) = 4 \] ### Step 3: Explanation of the Function The function \( f(x, y) \) effectively computes the average of \( x \) and \( y \) while adjusting for the distance between them. - When \( x \) is greater than \( y \), \( |x - y| \) adds to the sum, resulting in a value that is influenced more by the larger number. - Conversely, when \( y \) is greater than \( x \), the same effect occurs, but the function still averages the two values. In essence, this function can be interpreted as a way to find a "weighted average" of \( x \) and \( y \) that takes into account their difference, ensuring that the output is always at least as large as the larger of the two inputs. ### Summary of Results - For \( (3, 1) \), \( f(3, 1) = 3 \) - For \( (1, 4) \), \( f(1, 4) = 4 \) This function is useful in scenarios where the relationship between two variables is important, and it provides a way to balance their contributions based on their relative sizes.