Mod. 1 – Functions
Module 1 – Analyzing Functions
Review- Functions
A relation is a set of ______________________. Functions: A function is a
_________________ where each ____________ or x-coordinate has only one ______________
or y-coordinate paired with it.
How to test if a relation is a function: 1. Look at the x-values, _________________.
2. Use the Vertical-Line Test.
1.
(1, 1), (2, 2), (3, 3), (4, 4)
2.
(1, 1), (1, 2), (2, 3), (3, 4)
3.
4.
6.
8.
5.
7.
1.1 – Domain, Range, and End Behavior
The set of all possible _____________ in the relation form the domain. The set of all possible
__________________ in the relation form the range.
Domain and Range from Graphs
a)
When determining the domain, look at the graph from __________ to
__________.
b)

When determining the range, look at the graph from __________ to
__________.
Take note of any x- or y-values that are _______________________. Numbers not
used ______________________________ can't be included in the domain or range.
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Mod. 1 – Functions
End Behavior: describes what happened to f(x) as x approaches positive infinity and negative
infinity. In other words  what happens to the y values on the extreme left and right of the
graph. Do they increase forever, decrease forever, or approach a certain number?
Examples: Find the domain and range of each. State the end behavior.
(1)
(2)
(3)
(4)
(5)
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Mod. 1 – Functions
(6)
A grocery store stocks shelves with 100 cartons of strawberries before the store opens. For the
first 3 hours the store sells 20 cartons per hour. Over the next two hours no cartons are sold.
The store then restocks 10 cartons each hour for the next 2 hours. In the final hour the store is
open 30 cartons are sold.
1.2– Characteristics of Function Graphs
Rate of Change - The amount which a variable changes over a specific period. On a graph
average rate of change describes the amount the y-values change in comparison to the change in
the x-values. Sound familiar?? This is the same as the ______________!
Formula for Average Rate of Change:
or,
Ex) What is the average rate of change of the graph below over the interval 0 ≤ 𝑥 ≤ 100 ?
What does this mean given the context of the situation?
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Mod. 1 – Functions
Other Mathematical Vocabulary used to describe a graph:
What it looks like
How to describe it
mathematically
X – intercept
Root
Zeros of the function
Y- intercept
Maximum
Minimum
Positive
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Mod. 1 – Functions
Negative
Increasing
Decreasing
1.3– Transformations of Function Graphs
Transformations- There are numerous ways to apply transformations to a function to create a
new function. Such transformations include shifting the graph up, down, left, right, or flipping
it over one or both of the axis, or stretching it vertically or horizontally (or a combination of any
of the above).
Note: Use your calculator to compare the graphs of the original (most basic) function and the
new transformed function – focusing on specific points (like the max/min) can help
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Mod. 1 – Functions
A table of Basic Functions, with equations and graphs, is given below.
Equation
Graph
Equation
Graph
y  x2
y x
y x
ybx
y  x3
y3 x
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Mod. 1 – Functions
How Changing the Equation Effects the Graph
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Mod. 1 – Functions
Examples:
List the transformations applied to the original equation to arrive at the new equation.
1.
3.
Original:
New:
yx2
2.
𝑦 = (𝑥 + 1)2
Original:
y  x3
New:
y  x3  2
4.
Original:
New:
Original:
New:
y x
y x
y3 x
y  43 x
1
5. Describe the transformations on f(x) that are present
on the function 𝑓 (2 𝑥 − 3) + 5
6. Describe the transformations on f(x) that are present
on the function −4𝑓(𝑥 + 7) − 1
7. Original:
𝑓(𝑥) = 𝑥 2
Transformation: vertical shift 3 units up
8. Original:
𝑓(𝑥) = √𝑥
Transformations:
horizontal stretched by
a factor of 2 units
9. Original:
𝑓(𝑥) = 𝑥 3
Transformations:
reflection in x-axis and
shifted 2 units left
10. Original: 𝑓(𝑥) = |𝑥|
Transformations:
shifted up 3 units and
vertical stretch by a factor of 2
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Mod. 1 – Functions
Review- Evaluating Functions
Evaluating functions is as easy as pie! Substitute the input into the function in place of x
and then simplify the right side of the function.
If f(x) = 2x + 5. Find f (3).
If R ( y )  y 3  3 y 2  5 y  6 find 𝑅( −2)
If h( x)  2 x 2  6 x  3 find ℎ(4𝑎)
If f ( x)  x 2  2 x  1 find f (3h2)
Given, (𝑥) = 3𝑥 + 2 , for what value of x will
𝑓(𝑥) = 11 ?
For what domain will 𝑔(𝑡) = 8, if
𝑔(𝑡) = 3𝑡 − 4 ?
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Mod. 1 – Functions
Function Compositions: When two functions are connected in such a manner that the
__________ from one function becomes the _____________ for the next function it is referred
to as a composition of the two functions.
Notations used to represent a function compositions include:
_________________________________________
Examples: Evaluate the function below. HINT- Rewrite them in parenthesis notation & follow
PEMDAS.
If 𝑓(𝑥) = 𝑥 − 2 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 , 𝑓𝑖𝑛𝑑 (𝑓 ∘
𝑔)(−5)
If 𝑔(𝑥) = 2𝑥 𝑎𝑛𝑑 ℎ(𝑥) = 𝑥 2 + 4, 𝑓𝑖𝑛𝑑 (ℎ ∘
𝑔)(0)
If 𝑓(𝑥) = 2𝑥 2 + 𝑥 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 − 1, 𝑓𝑖𝑛𝑑 (𝑔 ∘
𝑓)(𝑤)
If 𝑓(𝑥) = √𝑥 − 1 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 +
1, 𝑓𝑖𝑛𝑑 (𝑓 ∘ 𝑔)(𝑥)
1.4 – Inverses of Functions
Inverse of a function: The inverse of a function is the set of order pairs obtained by reversing
the __________________________ of the original function. Graphically speaking, an inverse is
the _______________of a graph reflected over the line ______________.
Notation: If
is a given function, then______________________ denotes the
inverse.
Note: The inverse of a function may ________________be a function.
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Mod. 1 – Functions
To find 𝑓 −1 (𝑥) when given the equation….
1) Write the equation in __________
2) Switch the x and y variables. (Just the variables, not any “operations”)
3) Algebraically _____________________.
1) Write the equation for the inverse of
y  4x  8
1) If f ( x)  x 2  7 , find 𝑓 −1 (𝑥)
To graph 𝑓 −1 (𝑥) when given the graph of 𝑓(𝑥)….
1) Make a table of the key points on the given graph.
2) Swap ____________________ to create a table for 𝑓 −1 (𝑥)
3) Plot the new points and sketch the graph.
**Remember the inverse of a function if the reflection of the graph over the line
y=x!
Given the following graph, graph the inverse 𝑔−1 (𝑥)
Interesting Math fact - If a function is composed with its inverse, the result is the starting
value. ____________________________
This idea can be used to prove that two functions are inverses of each other!
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Mod. 1 – Functions
Use the following function 𝒇(𝒙) = 𝒙 + 𝟒 to solve the problems.
(f
f 1 )(4)
( f 1 f )(2)
Extra Practice –
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