Name:______________________________
Algebra 2
Objective:
Date:____________________________________
Lesson- Relations and functions
To know the definitions of relations and functions. To understand the difference between what
is a function and what is not. To be able to determine whether a relation is a function.
Definitions:
Relation-
Function-
Domain-
Range-
PBLM SET.
1.
State whether the relation is a function or not: Identify the Domain and Range.
a.
{(-2, 0), (3, 2), (4, 5)}
b.
2.
{(6, -2), (3, 4), (6, -6), (-3, 0)}
Which relation is a function? Why?
(a)
(b)
(c)
(d)
3.
Find the Domain and Range of each choice in exercise #2.
4.
Determine whether each of the following is a function. Justify your answer. Find the Domain and
Range of each.
a.
f ( x)  x  3
b.
f(x)  - x 2  2x - 27
1
2
3
4
5
6
7
Name:______________________________
Algebra 2
Objectives:
Date:____________________________________
Lesson- Linear Functions
To know the various properties of a linear function. To understand the processes for writing and
graphing various types of linear functions.
Do Now: State the four different types of slope and give an example for each:
Linear Function:
Forms of Linear Functions:
1.
slope-intercept form:
2.
standard form:
3.
point-slope form
Ex 1: Write the linear equation in slope intercept, standard, and point-slope form given that the line passes
through (5, 2) and (7, 9)
8
Ex 2: Write the equation of the horizontal line that passes through (-9, 2)
Parallel & Perpendicular Linear Function Rules:
Parallel
Perpendicular
Ex 3: Write the linear equation in standard form given that the line passes through (-2, 10) and is parallel to
the graph of y  3 x 
4
5
Ex 4: Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of
y
2
4
x
3
7
Intercepts
x-intercepts:
Ex 5: Find the x- and y- intercepts of f ( x) 
y-intercepts:
1
x  2 . Graph the linear function.
3
9
Name:______________________________
Algebra 2
Objectives:
Date:____________________________________
Lesson- Evaluating Functions
To know what it means to evaluate a function. To understand how (and be able) to evaluate a
function algebraically and graphically.
Notation for a function:
What does “evaluate a function” mean?
Evaluating Functions Algebraically
1. find f(-1) if f(x) = x2 – 1
2. find h(3) if h(x) = 3x2
3. find f(-7) if f(w) = 16 + 3w – w2
4. find g(m) if g(x) = 2x6 – 10x4 – x2 + 5
5. find k(w + 2) if k(x) = 3x + 4
6. find h(a – 2) if h(x) = 2x2 – x + 3
Evaluating Functions Graphically
1.
2.
3.
10
11
12
13
14
15
16
Name:______________________________
Algebra 2
Objectives:
Date:____________________________________
Lesson- One-to-One and Onto
To know what it means for a function to be One-to-one or Onto. To be able to distinguish
between One-to-one and Onto.
Definitions
Abscissa-
Ordinate-
One-to-one Functions
A function is one-to-one when no two ordered pairs in the function have the same ordinate and different
abscissas. The best way to check for one-to-oneness is to apply the vertical line test and the horizontal line
test. If it passes both, then the function is one-to-one. (**Note: if a function is not one-to-one, it does not have
an inverse**)
Onto Functions
A function is Onto if each ordinate associated with an abscissa. Multiple abscissas may map onto the same
ordinate. (**Note: if a function does not use all y-values in a Cartesian plane, it cannot be onto)
Examples:
Determine whether the following refers to a function one-to-one, onto, both or neither.
Explain your reasoning.
f ( x)  3  5 x
f ( x)  x 2  1
1)
2)
3)
f ( x)  2 x  1
17
4)
5)
6)
7)
8)
9)
18
Name:______________________________
Algebra 2
Date:____________________________________
Lesson- Composition & Inverse of Functions
Objective:
To know how to find the composition and inverse of a function. To understand the process for
finding the composition and inverse of a function. To be able to recognize an inverse
graphically.
Do Now:
Evaluate f ( x)  x3  x for x  2
Composition of Functions
“following” one function with another.
Notation:
Both of the following mean “f following g.”
f ( g ( x))
and
( f  g )( x)
Ex 1:
f ( x)  x  5
g ( x)  4 x
Find: a) f ( g ( x))
b) f (g (2))
c) ( g  f )(3)
d) g ( f ( x))
Would you say that a composition is a commutative operation? Why/why not?
Ex 2:
h( x )  x 2
r ( x)  x  3
Find: a) h( r ( x ))
b) r ( h( x ))
c) h(r (5))
19
Inverse Functions
Definition:
Steps:
1. Write the equation in terms of x and y.
2. Switch the x with the y.
3. Solve for y.
Ex 1: Find the inverse of y  4 x  8
Ex 2: Find the inverse of f ( x)  5 x  2
Ex 3: Find the inverse of g ( x)  x 2  4
Ex 4: Graph y  4 x  8 and it’s inverse on the axes below.
y
x
Ex 5: Looking at the graph of a line, can you find a way to graph it’s inverse?
20
Name:______________________________
Algebra 2
Date:____________________________________
Lesson- Operations with Functions
Operations with Functions: given functions f and g
sum:
difference:
product:
f  g(x)  f (x)  g(x)
f  g(x)  f (x)  g(x)
f  g(x)  f (x)  g(x)
f
f ( x)
quotient:  ( x ) 
, where g( x )  0
g( x )
 g
Given functions f and g: (a) perform each of the basic operations, (b) find the domain for each
(1) f ( x)  3 x  1; g ( x)  x
(2) f ( x)  5 x  4 ; g ( x)  x 2  1
(3) f ( x)  5  x ; g ( x) 
x 1
21
Name:______________________________
Algebra 2
Objectives:
Date:____________________________________
Lesson- Function transformations
To know the rules for various transformations such as: translations, reflections, symmetry,
rotations, and dilations. To understand the process for transforming coordinates, lines, and
curves. To be able to conduct various transformations and compositions of transformations.
f ( x)  x  1
Do Now: Sketch the graph the following polynomial:
y
x
Definitions:
Pre-Image:
Image:
Types of Transformations and their specific rules
1.
2.
3.
4.
22
23
24
25
26
27
28
Unit 2: Relations & Functions
Definitions, Properties & Formulas
Relation
a set of ordered pairs (x, y)
Domain
the set of all x-values of the ordered pairs
Range
the set of all y-values of the ordered pairs
Function
Slope
a relation in which each element of the domain is paired with exactly one
element in the range.
the slope, m, of the line through (x1, y1) and (x2, y2) is given by the
y  y1
following equation, if x1  x2: m  2
x 2  x1
y
Types of Slope
Positive
y
x
Negative
x
y-intercept
where the graph crosses the y-axis
x-intercept
where the graph crosses the x-axis
Slope-Intercept
Form
Standard
Form
Point-Slope
Form
Parallel
Lines
Perpendicular
Lines
y
y
x
Zero
horizontal line:
y=b
x
Undefined
vertical line: x
=a
y = mx + b
where m represents the slope and b represents the y-intercept of the
linear equation
Ax + By = C
where A, B, and C are constants and A  0 (positive, whole number)
y – y1 = m(x – x1)
where m represents the slope and (x1, y1) are the coordinates of a point
on the line of the linear equation
Two non-vertical lines in a plane are parallel if and only if their slopes are
equal and they have no points in common. (Two vertical lines are always
parallel.)
Two non-vertical lines in a plane are perpendicular if and only if their
slopes are negative reciprocals. (A horizontal and a vertical line are
always perpendicular.)
29
Vertical Line
Test (VLT)
Horizontal Line
Test (HLT)
One-to-One
Functions
Onto Functions
Inverse
Relations &
Functions
Writing Inverse
Functions
Operations with
Functions
If any vertical line passes through two or more points on the graph of a
relation, then it does not define a function.
If any horizontal line passes through two or more points on the graph of a
relation, then its inverse does not define a function.
a function where each range element has a unique domain element
(use HLT to determine)
All values of y are accounted for
f -1(x) is the inverse of f(x), but f -1(x) may not be a function
(use HLT to determine)
To find f -1(x):
(1) let f(x) = y
(2) switch the x and y variables
(3) solve for y
(4) let y = f -1(x)
sum:
(f + g)(x) = f(x) + g(x)
difference:
(f – g)(x) = f(x) – g(x)
product:
(f  g)(x) = f(x)  g(x)
quotient:
f
f ( x)
 ( x ) 
, where g( x )  0
g( x )
 g
Reflections: rx axis ( x, y)  ( x, y)
Dilations:
Transformations
ry axis ( x, y )  ( x, y )
rorigin ( x, y )  ( x, y )
Dk ( x, y)  (kx, ky)
Translations: Ta ,b ( x, y )  ( x  a, y  b)
Rotations:
R0,90 ( x, y )  ( y, x)
30
Name:________________________________
Algebra 2
Date:_________________________________
Review- Function test
Objective: To review the material that you will be tested on as part of Test #1-Functions. These topics are in
the outline below:
Functions
a. Identifying functions
b. Domain and Range of functions
c. Linear Function
i.
Finding x and y intercepts
ii.
Writing and graphing the equation of line in slope intercept form
iii.
Parallel and perpendicular lines and their graphs
d.
e.
f.
g.
h.
i.
j.
k.
Evaluating functions graphically
Evaluating functions algebraically
Identifying one-to-one functions
Identifying onto functions
Composition of functions
Inverse functions
Operations with Functions
Transformation of functions
Below you will find a sample of the types of problems you can expect to see on the test.
a.
Which graph of a relation is also a function?
(a)
b.
ci.
(b)
Determine the Domain and Range of:
f ( x)  3x  4
i.
(c)
(d)
ii.
g ( x)  x 2  9
Find the x and y intercepts for the following linear equations:
1. x  3 y  7
2. 3x  4 y  12
31
cii.
Write and graph the equation of the line given the following information:
1. m  3, and passes through (3,2)
2. passes through (5,1) and (2,0)
ciii.
1.
Write & graph the equation of the line that is parallel to y  3x  2 and passes through
(4,1).
2.
Write and graph the equation of the line that is perpendicular to y  3x  2 and passes
through its x intercept.
d.
If the following graph is y = f(x), what is the value of f(1)?
(a) -1
(b) -2
(c) 1
(d) 2
32
e.
f.
Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1)
Which function is not one to one?
(a)
g.
(b)
(c)
(d)
(c)
(d)
Which function is not onto?
(a)
(b)
h.
Given f ( x)  3x  4; g ( x)  x 2  9 , find ( f  g )( x) and ( g  f )( x) .
i.
Find the inverse of the following and state the domain.
a.
f(x) = 5x + 2
b.
f (x) 
4
x3
33
j.
Perform the four basic operations on f ( x)  3x  4; g ( x)  x 2  9 and determine the domain of
the result.
k.
Complete the following transformations on graph paper. Label your images.
a.
rx axis (2,1)
b.
D2 [ f ( x)  x 2  1]
c.
R0,90 [ g ( x)  2 x  1]
d.
T2,3 [h( x)  2 x 2  4 x  2]
a.
b.
y
y
x
c.
x
d.
y
x
y
x
34