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

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1. Let $$ x = 3 $$ and $$ y = 5 $$. Then,  
   $$
   |x-y| = |3-5| = 2.
   $$
   Substituting in the function:
   $$
   f(3, 5) = \frac{1}{2}(3+5+2) = \frac{1}{2}(10) = 5.
   $$
   Since $$\max(3,5)=5$$, the function returns the maximum value.

2. Let $$ x = 7 $$ and $$ y = -2 $$. Then,  
   $$
   |x-y| = |7-(-2)| = |7+2| = 9.
   $$
   Substituting in the function:
   $$
   f(7, -2) = \frac{1}{2}(7+(-2)+9) = \frac{1}{2}(14) = 7.
   $$
   Since $$\max(7,-2)=7$$, the function again returns the maximum value.

In general, when $$x \geq y$$, we have $$|x-y| = x-y$$ so that
$$
f(x, y) = \frac{1}{2}(x+y+x-y) = x.
$$
Conversely, when $$x \leq y$$, $$|x-y| = y-x$$ and
$$
f(x, y) = \frac{1}{2}(x+y+y-x) = y.
$$
Thus, the function returns the maximum of $$x$$ 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, when $$ x = 3 $$ and $$ y = 5 $$, $$ f(3, 5) = 5 $$, and when $$ x = 7 $$ and $$ y = -2 $$, $$ f(7, -2) = 7 $$.

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

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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 of and . Show two of the calculations below and briefly explain what this function does. to