SECTION 3 STUDY GUIDE
By Crista Cappuccilli
TABLE OF CONTENTS
How can you identify and represent functions?
How can you represent name and solve a function?
How do you add and subtract functions?
How do you multiply functions?
How do you divide functions?
How do you apply the closure property when adding subtracting and
multiplying polynomials
How do you know if a graph is a function ?
How do identify key features of functions on graphs?
How do you determine the domain and range in piecewise functions?
How do functions transform?
VIDEO 1: HOW DO YOU IDENTIFY AND REPRESENT FUNCTIONS
Vocabulary
Domain- The set of values
of X used for the input of the
function.
Range-The set of values
calculated from the domain for
the output of the function.
Functions is when you take an input value and
assign it 1 output value. The input value CANNOT
have more than 1 function. However the output
values can have more then 1 input value. You can
display the functions in many ways. Including
1.graph, 2.table, 3.ordered pairs. Your input, X,
then your function, f(x). The function will give you
your output.
Examples:
Nick earns $5 each day to
mow a lawn. Explain this in
a function.
Input: Number of days
Output: Money earned
Function: f(x)=$5x
**Inputs can only have one output, but
outputs can have more than 1 input!
VIDEO 2: HOW CAN YOU REPRESENT, NAME, AND SOLVE A FUNCTION
You can represent, name , and solve a function by plugging in a value to replace the
variable with, then you can solve the equation finishing with an output value.
Example 1:
Example 2: Table
This table consists of three parts, the input,
output, and function to gain the output. The
function for this problem is f(x)= x^24. If
we plug in a value for x and follow the order of
operations in this function, we will end up with
an output. The table to the right let's us easily
see what is happening.
VIDEO 3: HOW DO YOU ADD AND SUBTRACT FUNCTIONS?
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Subtraction Example:
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Addition Example:
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When adding functions there is no special trick. You do not need the ( )s because you
are adding like terms between however many functions you have. However with
subtracting, you keep the ( )s for one extra step. You place a subtraction sign in front
of the 2nd function. Then you distribute the subtraction sign into each individual
part of the function. This turns the terms sign opposite. For example if it was
positive it is now negative and vise versa. When you have distributed the subtraction
sign you can now get rid of the ( )s. Now you combine like terms like you did in
addition. And then the final step to both adding and subtracting is writing in
standard form. The highest exponent goes first and so on. Then your done!
VIDEO 4: HOW DO YOU MULTIPLY FUNCTIONS
You can multiply functions with two different ways, the box method and the
distributive property. Either will get you the accurate answer, however if you
are multiplying more than two functions, I suggest the box method.
Box Method
1. Draw a box, the amount of rows and
columns you draw vary of the terms in
your function.
How To
2. Label the rows and columns, the top and
side represent your different equations.
3. Multiply across pitting your answer in the
corresponding box.
4.
Combine like terms and write in
K
Multiply
standard form.
Example:
Functions
To distribute most of the time you use
( )s, you "pass out" the number in the
outside to the numbers on the inside.
This way your multiplying is evenly
spread throughout the problem. A
more visual way to do this is the box
method.
Distributive Property
VIDEO 5: HOW DO YOU DIVIDE FUNCTIONS?
To divide functions you have to find the
constraints. When dividing functions you
look for solutions to the dominator function,
but make sure the bottom function will not
equal 0. Also when dividing functions you
want to be able to rewrite the fraction into
multiplication. If your solving with two
different functions in fraction form, cross
multiply .
Constraint
Vocabulary
Constraints: Values
of the variable that
make the dominator
equal to zero.
VIDEO 6: HOW DO YOU APPLY THE CLOSURE PROPERTY WHEN
ADDING, SUBTRACTING, AND MULTIPLYING POLYNOMIALS?
What is the Closure Property?
"A set is closed for a specific operation if
Closed
Polynomials
are closed under
addition
Polynomials
are closed under
subtraction
Polynomials
are closed under
multiplication
and only if the operation on two
elements of the set always
produces an element of the
Not
Closed
same set."
Polynomials are not
closed under division
Integers are not closed
under division
The closure property is when the
numbers you are using to add, multiply,
or subtract in a polynomial will always
end up the same whether it rational or
irrational. If you divide, you can end up
with rational or irrational numbers, so
it is not closed.
VIDEO 7: HOW DO YOU KNOW IF A GRAPH IS AN FUNCTION?
Fails the
Vertical Line
Test
Passes the
Vertical Line
Test
X
You know if a graph is a function by performing the vertical line test. If a
vertical line passes through the graphed line more that once it is not a
function.
Passes the
Vertical Line
Test
Fails the
Vertical Line
Test
VIDEO 8: HOW DO IDENTIFY KEY FEATURES OF FUNCTIONS ON GRAPHS?
Domain
Range
Definition: The input or
the X values
Definition: The output or the
Y values
Decreasing Intervals Relative Maximum
Definition: As the x-values
increase the y-values decrease
Definition: The point on the
graph where the interval
changes from increasing no t
decreasing/ the highest point
on the graph
Increasing Intervals
Definition: As the x-values
increase the y-values
increase
Relative Minimum
Definition: The point on the
graph where the interval
changes from decreasing to
increasing/ the lowest point
on the graph
VIDEO 9: HOW DO YOU DETERMINE THE DOMAIN AND RANGE IN
PIECEWISE FUNCTIONS?
You write your domain and range as a interval. Like in the example below. This
only applies to piecewise functions.
VIDEO 10: HOW DO FUNCTIONS TRANSFORM?
Vertical
Moves up or down form
parent function.
If adding positive numbers
the transformation moves
up, if subtracting negative
numbers, the transformation
is down
Functions transform in 2
different ways, vertical, up or
down, and horizontal, right or
left. These transformations are
added on to the parent function
and then graphed. Here are
examples.
Horizontal
Moves right or left from the
parent function
If adding a positive number,
transformation moves left. If
subtracting transformation
moves right.
