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 place $$ x $$ and $$ y $$. Show two of the calculations below and briefly explain what this function does.

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1. Calculation for $$ x = 5 $$ and $$ y = 3 $$:

- Compute the absolute difference:  
     $$ |5 - 3| = 2 $$
     
   - Substitute into the function:  
     $$ f(5, 3) = \frac{1}{2}\left(5 + 3 + 2\right) = \frac{1}{2}(10) = 5 $$
     
   - Since $$ 5 $$ is the larger number, the function returns $$ 5 $$.

2. Calculation for $$ x = -4 $$ and $$ y = 2 $$:

- Compute the absolute difference:  
     $$ |-4 - 2| = |-6| = 6 $$
     
   - Substitute into the function:  
     $$ f(-4, 2) = \frac{1}{2}\left(-4 + 2 + 6\right) = \frac{1}{2}(4) = 2 $$
     
   - Here, the larger number is $$ 2 $$, and the function returns $$ 2 $$.

**Explanation:**  
In both cases, after simplifying the expression, the function $$ f(x, y) = \frac{1}{2}(x+y+|x-y|) $$ always returns the greater of $$ x $$ and $$ y $$. This is because when $$ x \ge y $$:
$$
|x-y| = x-y \quad \Longrightarrow \quad f(x,y)=\frac{1}{2}(x+y+x-y)=x,
$$
and when $$ y > x $$:
$$
|x-y| = y-x \quad \Longrightarrow \quad f(x,y)=\frac{1}{2}(x+y+y-x)=y.
$$
Thus, the function effectively computes the maximum of the two values.
The function $$ f(x, y) = \frac{1}{2}(x + y + |x - y|) $$ always returns the larger of the two numbers $$ x $$ and $$ y $$. For example, when $$ x = 5 $$ and $$ y = 3 $$, $$ f(5, 3) = 5 $$, and when $$ x = -4 $$ and $$ y = 2 $$, $$ f(-4, 2) = 2 $$.

(extra credit 10 pts.) Consider the function of two variables given by
Experiment with on scratch paper for various real-number inputs in place
and . Show two of the calculations below and briefly explain what this function does.

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(extra credit 10 pts.) Consider the function of two variables given by Experiment with on scratch paper for various real-number inputs in place and . Show two of the calculations below and briefly explain what this function does.