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\noindent
{\Large 
FUNCTIONS WORKSHEET \hfill Name:\underline{\hspace{2.8in}}\\ \\
MA~202 \\
Spring Semester 2004}
\vspace{.25in}

\noindent
{\bf WARNING:}  You must {\bf SHOW ALL OF YOUR WORK}.  You will receive 
NO CREDIT if you do not show your work.

\noindent
{\bf DUE:} Thursday, 05 February 2003 
\begin{enumerate} 
\item Jake is is $y$ years old.  Write an algebraic expression with the
variable $y$ for Joni's age if:
	\begin{enumerate}
	\item Joni is three years older than Jake was five years ago.
	\item Joni is twice as old as Jake will be in six years.
	\item Joni is half as old as Jake's mother who is three times as
		old as Jake was four years ago.
	\end{enumerate}

\item Consider the four number machines shown below. 
	\begin{center}
	\epsfysize 1.5in
	\epsffile{pictures/machine1.eps}
	\end{center}
Machine A takes a number $x$ and returns $x+3$.  For example, if we place
the number $5$ in machine $A$, it will return the number $8$.  If we place
the number $-1.2$ in machine $A$, it will return the number $1.8$. Machines
B, C, and D act similarly.  If we place the number $4$ in machine $B$, it
will return the number \spc.  If we place the number $-7$ in machine $C$,
it will return the number \spc.  If we place the number $5.2$ in machine
$D$, it will return the number \spc.

Machines A, B, C, and D are {\em simple machines} because they only perform
one operation on a number.  Let's take a look at a machine which is not a
simple machine.  Consider machine E which is shown below.
	\begin{center}
	\epsfxsize 1in
	\epsffile{pictures/machine2.eps}
	\end{center}
Machine E is a {\em complex machine} because it performs more than one
operation on the number that is input.  If we place the number $x$ into
machine E, it will first add three to $x$ and then square $(x+3)$ yielding
$(x+3)^2$.  We can construct E from simple machines.  For example, we can 
input a number into machine A and then place the output from machine A into 
machine B.  When machines A and B are linked in this way, together they perform
the same operations that machine E performs. 
	\begin{center}
	\epsfxsize 2in
	\epsffile{pictures/machine3.eps}
	\end{center}

I have described several complex machines below.  You need to construct
each complex machine from the simple machines A, B, C, and D.  Your answer
should be a diagram similar to the ones shown above.
	\begin{itemize}
	\item When the number $x$ is placed in machine $F$, it returns
	$\bfrac{(x+3)^2}{2}$.
	\item When the number $x$ is placed in machine $G$, it returns
	$\bfrac{x^2+3}{2}$.
	\item When the number $x$ is placed in machine $H$, it returns
	$5x+3$.
	\item When the number $x$ is placed in machine $I$, it returns
	$5(x+3)$.
	\item When the number $x$ is placed in machine $J$, it returns
	$\bfrac{(5x+3)^2}{2}$.
	\end{itemize}

Mathematicians are, by nature, somewhat lazy when it comes to writing.  
We really like shorthand --- provided that it is used correctly.  (Remember
that you need to be very, very careful with the = symbol.  You may only use
it to mean that quantities are, in fact, equal.  You may not use this
symbol for the word ``implies.'')  It requires a lot of time to draw number
machines, especially if you are using a computer as I have been doing to
write this worksheet.  The shorthand for number machines is called {\em
function notation}.  Here is how it works.  Consider number machine A.  The
name of this machine is ``A.''  It takes the input $x$ and returns the
value $x+3$.  Therefore the shorthand for machine A is $A(x)=x+3$.  Since
$A(x)=x+3$, we can either say that the output of machine A is $x+3$ or
$A(x)$.  If we
want to put a specific number into the machine we can replace the input
variable $x$ with our number.  For example, if we put the number $5$ into
machine A, it will return the value $8$.  We represent this by $A(5)=8$.
Notice that $A(5) = 5+3$, that is $5$ replaces {\em all} of the $x$ values
in our notation.
	\begin{itemize}
	\item Write the function notation for machines B--J.
	\item What is $J(2)$?
	\item What is $C(-5)$?
	\item What is $D(4)$?
	\item What is $I(x^2+h)$?
	\item What is $B(x+3)$?
	\item What is $B(A(x))$?
	\end{itemize}
Recall that we can construct machines E--J from our simple machines.  How
would we represent this using function notation?  Let's take a look at
machine E.  We constructed E from simple machines by joining machine A to
machine B.  We input the value $x$ into machine A and it returns the value
$A(x)$.  Then this value goes into machine B, that is $A(x)$ goes into
machine B.  Therefore B should return the value $B(A(x))$.  Hence, we can
write $E(x) = B(A(x))$.  This function notation is used to show how the
complex machine E can be constructed from the simple machines A and B.
	\begin{itemize}
	\item Each of the complex machines F--J can be constructed from the 
		simple machines A--D.  Describe these constructions using
		function notation.  
	\end{itemize}

Now let's see how we can use these machines to help us solve equations.
Suppose we want to solve \[(x+3)^2 = 5.\]  Notice that all the $x$'s are on
one side of the equation.  Also notice that the left hand side of the
equation is $E(x)$.  To solve this equation, we need to walk backwards
through the simple machine construction.  Remember, that we constructed E
from simple machines by sending a number through machine A and then through
B.  So if we are going to walk backwards through this construction we will
need to undo the operation in B and then undo the operation in A.  Since B
squares a number, we will need to undo this operation by taking the square
root of both sides of the equation.  Note that we will have to be careful
here to include both the positive and negative square root. This gives:
\begin{eqnarray*}
(x+3)^2 & = & 5 \\
\sqrt{(x+3)^2} & = & \sqrt{5} \\
x+3 & = & \pm \sqrt{5} \\
\end{eqnarray*}
Now we need to undo the operation of machine A.  Since A adds three to a
number, we will need to undo this operation by subtracting three from both
sides of the equation.  This gives:
\begin{eqnarray*}
(x+3)^2 & = & 5 \\
\sqrt{(x+3)^2} & = & \sqrt{5} \\
x+3 & = & \pm \sqrt{5} \\
x+3-3 & = & \pm \sqrt{5}-3 \\
x & = & \pm \sqrt{5}-3 \\
\end{eqnarray*}
	\begin{itemize}
	\item Verify that $\sqrt{5}-3$ and $-\sqrt{5}-3$ are both solutions
	to the equation $(x+3)^2 = 5$.
	\end{itemize}
We can use this process of walking backwards through simple machines to
solve lots of equations.  Solve the following equations.  Explain to your
group members how your solutions are related to walking backwards through
simple machines.
	\begin{itemize}
	\item $16 = \bfrac{(x+3)^2}{2}$.
	\item $\bfrac{1}{2} = \bfrac{x^2+3}{2}$.
	\item $x + 4.2 = 6x+3$.
	\item $5(x+3) = 7$.
	\item $\bfrac{(5x+3)^2}{2} = \frac{1}{3}$.
	\end{itemize}

\item For each complex function described below, describe the simple
machines that are needed to construct the complex function.  Then construct
the complex machine from the simple machines you described.  
	\be
	\item $k(x) = \frac{2x+3}{5}$
	\item $l(x) = \frac{1}{x+2}$
	\item $m(x) = \sqrt{x^2 - 1}$
	\item $n(x) = 3\sqrt{4x-5} - 7$
	\ee

\item Find the solution set for each equation.  Be sure to show all of your 
work and check your answers.
	\be
	\item $4x + 6 = 9x -2$
	\item $\frac{2x+3}{5} = \frac{1}{10}$
	\item $\frac{1}{x+2} = 7$
	\item $\frac{1}{x+2} = 0$
	\item $\sqrt{x^2 - 1} = 0$
	\item $3\sqrt{4x-5} - 7 = -1$
	\ee

\item Which of the following equations are identities?  Which are
conditional equations?  Which equations do not have real number solutions?
	\be
	\item $x+2 = 53$
	\item $2x + x^2 = x(2+x)$
	\item $(x + y)^2 = x^2 + y^2$ ({\em Be careful here.})
	\item $x^2 = \displaystyle{\frac{1}{\frac{1}{x^2}}}$
	\item $x^2 + 1 = 0$. 
	\item $\sqrt{x^2 - 1} = 0$
	\ee
\item Consider the sequence of polygon trains shown below.  
Write an algebraic expression using the variable $n$ to represent the number 
of edges in the $n^{th}$ polygon train.
\begin{center}
\epsfbox{pictures/polygontrain.eps}
\end{center}
\item Joni can mow Grandma's yard in 4 hours.  Jake can mow Grandma's yard
in 6 hours.   How long will it take Jake and Joni to mow Grandma's yard if
they work together?
\item Amy can make seven necklaces in one hour.  Katie can make thirteen 
necklaces in two hours.  How long will it take Amy and Katie to make fifty
necklaces if they work together?
\item Consider the function $y=f(x)$.  
	\be
	\item {\bf Circle the correct responses.}
		Suppose we want to graph this function in the Cartesian plane.
		The $x$ values represent the (input, output) of the
		function.  The $y$ values represent the (input, output) of
		the function.
	\item  Describe the vertical line test.  Explain why the vertical line 
		test works when we consider functions of the form $y =
		f(x)$.
	\item How would you have to adjust the vertical line test if we
	 	wished to consider functions of the form $x = f(y)$.	
	\ee
\item 
	\be
	\item Write a function $A(r)$ for the area of a circle of radius
		$r$.
	\item Write a function $B(d)$ for the area of a circle of radius
		$d$.
	\item Find $A(2)$, $A(4)$, $A(3)$, $A(6)$, $A(.5)$, and $A(1)$.
		What do each of these values represent?  
	\item How does the area of a circle change when we double its
		radius?  Verify your answer using function notation. ({\em
		Hint:}  If we double the radius $r$, then the new radius
		is expressed by \spc.  Input the new radius into your area
		function $A$.)
	\item How does the area of a circle change when we double its
		diameter.  Verify your answer using function notation.
	\item Which is the better deal: three ten inch pizzas for $\$20$ or
		one twenty inch pizza for $\$20$?
	\item Write a function $P(r)$ for the perimeter of a circle with
		radius $r$.
	\item How does the perimeter of a circle change when we double its
		radius?  Verify your answer using function notation.
	\ee
\item Use the Pythagorean Theorem to find the distance between $(4,5)$ and
$(7,16)$.
\item What do points on a horizontal line have in common?
\item What do points on a vertical line have in common?
\item Write an equation for the horizontal line through the point $(12,9)$.
\item Write an equation for the vertical line through the point $(12,9)$.
\item Consider the graph of the parabola given by $f(x) = x^2$.  
	\be
	\item Find the slope of the line through the points of the
		parabola at $x=2$ and $x=3$.
	\item Find the slope of the line through the points of the
		parabola at $x=-2$ and $x=-3$.
	\item How would your results for this problem be different
		if $f(x)$ had described a line?
	\ee
\item Write an equation for a line which passes through the point $(4,-1)$
	and has slope $6$.
\item Write an equation for the line which passes through the points
	$(-1,-2)$ and $(5,6)$.
\item Write an equation for the line which passes through the points
	$(-1,-2)$ and $(-1,6)$.
\item Write an equation for the line which passes through the points
	$(-1,6)$ and $(5,6)$.
\item Write an equation for the line which passes through $(-4,-5)$ and is
	parallel to the line $4x = 6+9y$.
\item Read Sections 8.1--8.3 of your text.
\ee

\end {document}
http://www.olemiss.edu/mathed/middle/fraction.htm
