7b - Lab #7 Piecewise fn's and Operations on fn's.sagews

##Lab 7 Piecewise fn's and Operations with fn's

We'll start by looking at the syntax for defining piecewise functions using the builtin function called Piecewise. Let's have a 2-piece(can go with more) over the overall interval, $[-5,5]$. We want the function to return $4-x$ over the interval $-5<x< 1$ and $x^2-1$ over the interval $1\le x<5$. The function's name will be pw. One important thing to mention!! SAGE doesn't allow half-open or half-closed intervals! To it, the intervals are always closed.

x=var('x')
pw=Piecewise([[(-5,1),4-x],[(1,5),x^2-1]],x)#take careful note how this is typed!!!
show(pw)
pw(1)#This is what I was talking about in red above!  SAGE returns the average of the two outputs for x=1
pw.plot()#another way to create a quick plot
︠b9ef9790-0f28-428b-b13b-af99b7bc5d30i︠
%md
Next we look at operations on functions and function composition.  I will use the two functions, $f(x)=2x+3\mbox{ and }g(x)=3x-7$.  Step 3 below shows the two functions, their sum, difference, product and quotient.  It also shows the composition, $f(g(x))$.

Next we look at operations on functions and function composition. I will use the two functions, $f(x)=2x+3\mbox{ and }g(x)=3x-7$. Step 3 below shows the two functions, their sum, difference, product and quotient. It also shows the composition, $f(g(x))$.

x=var('x')
f=2*x+3; g=3*x-7
show([f,g,f+g,f-g,expand(f*g),f/g,f(g)])
︠38607fec-633e-45b7-a2b5-ebf36f67a0a6is︠
%md
Your turn. . . Let $f(x)=\sqrt[3]{6-3x}$ and $g(x)=2x^3+1$
Find the following all together or one at a time:
1.  $f+g$
2.  $f-g$
3.  $fg$
4.  $f/g$(determine its domain)
5.  $f(g)$
6.  $g(f)$
Evaluate all results for $x=2$

Your turn. . . Let $f(x)=\sqrt[3]{6-3x}$ and $g(x)=2x^3+1$ Find the following all together or one at a time:

1. $f+g$

2. $f-g$

3. $fg$

4. $f/g$(determine its domain)

5. $f(g)$

6. $g(f)$ Evaluate all results for $x=2$

︠42f0f6f7-df19-420a-b942-b1b8e3b7cb07i︠
%md
Let's look at one more new concept in this lab.  It's called a **difference quotient**.  This idea is fundamental to the study of beginning calculus.  The difference quotient is a new function whose variable is $h$ that's built using an ordinary function of $x$.  Here I name it $diff\_quo$.  What $h$ represents is not important at this point.  In the example below,  my $f(x)=2x^2+3$.
Use my block to fashion code to calculate the difference quotients for the following functions:
1.  $\sqrt{x}$
2.  $x^3$
3.  $1/x$
4.  $x^2-5x+6$

Let's look at one more new concept in this lab. It's called a difference quotient. This idea is fundamental to the study of beginning calculus. The difference quotient is a new function whose variable is $h$ that's built using an ordinary function of $x$. Here I name it $diff\_quo$. What $h$ represents is not important at this point. In the example below, my $f(x)=2x^2+3$. Use my block to fashion code to calculate the difference quotients for the following functions:

1. $\sqrt{x}$

2. $x^3$

3. $1/x$

4. $x^2-5x+6$

h=var('h')
f(x)=2*x^2+3
diff_quo(h)=(f(x+h)-f(x))/h
diff_quo(h)

2*((h + x)^2 - x^2)/h