Math $$ 126 \quad $$ Written Homework 11 (extra credit) If you choose tó not do these problems, please do not turn in the assignment. 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 o $$ x $$ and $$ y $$. Show two of the calculations below and briefly explain what this function does.

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1. The given function is 
$$ f(x,y)=\frac{1}{2}(x+y+|x-y|). $$

2. To understand the behavior of $$ f $$, consider two cases based on the relationship between $$ x $$ and $$ y $$.

- **Case 1: $$ x \geq y $$**
     
     When $$ x \geq y $$, we have 
     $$ |x-y| = x-y. $$
     Substituting into the function:
     $$
     f(x,y)=\frac{1}{2}(x+y+(x-y))=\frac{1}{2}(2x)=x.
     $$

- **Case 2: $$ x < y $$**
     
     When $$ x < y $$, we have 
     $$ |x-y| = y-x. $$
     Substituting into the function:
     $$
     f(x,y)=\frac{1}{2}(x+y+(y-x))=\frac{1}{2}(2y)=y.
     $$

3. From these calculations, we see that:
   $$
   f(x, y) = 
   \begin{cases} 
   x & \text{if } x \geq y, \
   y & \text{if } x < y.
   \end{cases}
   $$
   In other words, $$ f(x,y) $$ returns the maximum of $$ x $$ and $$ y $$.

4. **Sample Calculations:**

- **Example 1:**
     Let $$ x=3 $$ and $$ y=1 $$. Since $$ 3 \geq 1 $$,
     $$
     f(3,1)=3.
     $$

- **Example 2:**
     Let $$ x=2 $$ and $$ y=5 $$. Since $$ 2 < 5 $$,
     $$
     f(2,5)=5.
     $$

Thus, the function $$ f(x, y)=\frac{1}{2}(x+y+|x-y|) $$ calculates the maximum of the two inputs $$ x $$ and $$ y $$.
The function $$ f(x, y) = \frac{1}{2}(x + y + |x - y|) $$ returns the larger of the two numbers $$ x $$ and $$ y $$. For example, if $$ x = 3 $$ and $$ y = 1 $$, $$ f(3,1) = 3 $$, and if $$ x = 2 $$ and $$ y = 5 $$, $$ f(2,5) = 5 $$.

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