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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Discussion 2 -- ``New Functions From Old Ones''}
\centerline{September 13, 2005}
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{\it Background}
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We have introduced the idea of shifting and stretching graphs
to produce new graphs.  Recall the idea:  In $y = f(x)$,
if we replace $y$ by $y-k$, then the graph is shifted up if $k>0$
and down if $k < 0$.  Similarly, if we replace $x$ by $x - h$, then
the graph is shifted right if $h > 0$ and left if $h < 0$.  
Replacing $f(x)$ by $cf(x)$ stretches the graph vertically if $|c| > 1$,
shrinks it if $|c| < 1$, and also reflects the graph across the 
$x$ axis if $c < 0$.  

We have also discussed function composition and inverse functions.
Today, we want to practice using some of these ideas.

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{\it Discussion Questions}
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A)  Sketch the graphs $y = f(x-3) + 1$ and $y = -2f(x+2)$ for
$f(x)$ given by the graph below.  Only show the portion of the
graph for the interval of $x$-values $-2 \le x \le 2$ in each case.

\vglue 1.5truein

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B)  Let $f(x) = -3x + 5$, $g(x) = {1\over x^2 + x}$.
What are the composite functions $f(g(x))$ and $g(f(x))$?
Give the formulas and simplify.  

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C)  Let $f(x)$ be the temperature in degrees Celsius when 
the mercury in a particular thermometer is $x$ centimeters long.  
Explain in practical terms the meaning of the value $f(10)$.  Why 
would we expect $f$ to be invertible?  
Explain in practical terms the meaning of the value $f^{-1}(30)$.


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D)  You are the owner of a taqueria with a very simple menu: just
one type of burrito.  You have fixed costs
of $\$200$ per day (rental of the shop, electricity, 
wages for your cashier).  The ingredients for one burrito cost
$\$4$.  If you sell $x$ burritos per day, what is your total
cost function $C(x)$?  Your function should be invertible.
Find a formula for the inverse function and explain in 
practical terms what it tells you.

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{\it Assignment}
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Group writeups due in class tomorrow, Wednesday, September 14.
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