Algebra 2H
Lesson- Relations and functions
Name:____________________________________
Date:_____________________________________
Objective:
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
Algebra 2H
Lesson- Linear Functions
Name:____________________________________
Date:_____________________________________
Objectives:
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)
4
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
4
the graph of y  3x 
5
Ex 4: Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of y 
Intercepts
x-intercepts:
2
4
x
3
7
y-intercepts:
1
Ex 5: Find the x- and y- intercepts of f ( x)  x  2 . Graph the linear function.
3
5
Algebra 2H
Lesson- Evaluating Functions
Name:____________________________________
Date:_____________________________________
Objectives:
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.
6
Algebra 2H
Lesson- Graphing Absolute Value Functions
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to graph a piecewise and absolute value function
Do Now:
State the domain for f ( x) 
x2 1
x
__________________________________________________________________________________________
Absolute Value Functions
y
Graph the Following
f ( x)  x
x
Now graph each of the following and discuss how each relates to f (x) from above.
g ( x)  2 x
h( x)  2 x  3
i ( x)  2 x  3
y
y
x
y
x
x
7
Algebra 2H
Solving Absolute Value Equations
Name:____________________________________
Date:_____________________________________
Objectives: To learn to solve absolute value equations and absolute inequalities.
Absolute Value Equations
ax  b  c
To solve ax  b  c create 2 equations 
and solve each.
ax  b  c
Example:
3x  1  2
Practice:
a. x  1  4
b. 3  y  5
c. 2  3d  4
d . 2m  1  2
8
Algebra 2H
Lesson- Solving Absolute Value Inequalities
Name:____________________________________
Date:_____________________________________
Objectives: To learn to solve absolute value inequalities.
Absolute Value Inequalities
There are three absolute value situations:
Case 1
ax  b  c
Case 2
ax  b  c
Case 3
ax  b  c
ax  b  c
ax  b  c
 c  ax  b  c
Either ax  b  c or ax  b  c
Examples:
a.
3x  1  2
b.
3x  1  2
c.
3x  1  2
d.
3x  1  2
Practice:
a. x  1  4
b. 3  y  5
c. 2  3d  4
d . 2m  1  2
9
10
Algebra 2H
Name:____________________________________
WKST- Mixed equation/inequality and absolute value set
Date:_____________________________________
Answer each of the following neatly and completely in the space provided.
Solve and graph each inequality:
1.
x  7x  6
2.
x  3  3(2 x  1)
3.
x3  4
4.
2x  5  x  1
5.
2x  3  5
6.
2 x  8
7.
x 2  2 x  24  0
8.
x 2  10 x  1  0
11
9.
x2  4  0
10.
x2  8x  7  0
11.
3x 2  10 x  8
12.
5d  7  28
13. Explain why the solution set of
x 2  9  0 is all real numbers.
14. Explain why the solution set of
x 2  16  0 is empty.
12
Algebra 2H
Lesson- Graphing Piecewise Functions
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to graph piecewise functions.
Do Now: Graph: f ( x)  3x  2 for  3  x  0
y
x
What is a piecewise function?
Graph the following:
2 x if x  0
f ( x)  
2 if x  0
y
x
2 x  1 if x  0
g ( x)   2
 x if x  0
y
x
13
14
Algebra 2H
Name:__________________________________
Mixed Wkst: Graphing Absolute and Piecewise Functions
Date:___________________________________
y
1. Graph the following function:
x  2 if x  1
f ( x)  
 x  2 if x  1
x
2. Graph each function, and state the domain and range
(1)
f ( x)  x  3
(2)
f ( x)  3 x  2
y
y
x
x
15
Algebra 2H
Lesson- One-to-One and Onto
Name:__________________________________
Date:___________________________________
Objectives:
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)
16
Examples:
Determine whether the following refers to a function one-to-one, onto, both or neither.
Explain your reasoning.
f ( x)  3  5 x
1)
2)
3)
f ( x)  x 2  1
f ( x)  2 x  1
4)
5)
6)
7)
8)
9)
17
Algebra 2H
Lesson- Composition & Inverse of Functions
Name:__________________________________
Date:___________________________________
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))
18
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?
19
Algebra 2H
Lesson- Operations with Functions
Name:__________________________________
Date:___________________________________
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)  3x  1; g ( x)  x
(2) f ( x)  5x  4 ; g ( x)  x 2  1
(3) f ( x)  5  x ; g ( x)  x  1
20
Algebra 2H
Lesson- Function transformations
Name:__________________________________
Date:___________________________________
Objectives:
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.
Do Now: Sketch the graph the following polynomial:
f ( x)  x  1
y
x
Definitions:
Pre-Image:
Image:
Types of Transformations and their specific rules
21
Extra Space
22
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 following
y  y1
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
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
Parallel Lines
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.)
Perpendicular
Lines
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.)
23
Vertical Line Test
(VLT)
Horizontal Line
Test (HLT)
One-to-One
Functions
Onto Functions
Inverse Relations
& Functions
Writing Inverse
Functions
Operations with
Functions
Transformations
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:
Dk ( x, y)  (kx, ky)
ry axis ( x, y )  ( x, y )
rorigin ( x, y )  ( x, y )
Translations: Ta ,b ( x, y )  ( x  a, y  b)
Rotations:
R0,90 ( x, y )  ( y, x)
24
Algebra 2H
Review- Function test
Name:__________________________________
Date:___________________________________
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. Evaluating functions graphically
e. Evaluating functions algebraically
f. Absolute Value Functions
g. Piecewise Functions
h. Identifying one-to-one functions
i. Identifying onto functions
j. Composition of functions
k. Inverse functions
l. Operations with Functions
m. 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
25
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
26
e.
Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1)
f.
What are the significance of the a,h,k values in the standard form of an absolute value function?
g.
Write f ( x)  2 x  2  2
h.
2
Which function is not one to one?
(a)
i.
(c)
(d)
(c)
(d)
Which function is not onto?
(a)
j.
(b)
(b)
Given f ( x)  3x  4; g ( x)  x 2  9 , find ( f  g )( x) and ( g  f )( x) .
27
k.
Find the inverse of the following and state the domain.
a.
f(x) = 5x + 2
b.
f (x) 
4
x3
l.
Perform the four basic operations on f ( x)  3x  4; g ( x)  x 2  9 and determine the domain of the
result.
m.
Complete the following transformations on graph paper. Label your images.
a.
b.
D2 [ f ( x)  x 2  1]
rx axis (2,1)
R0,90 [ g ( x)  2 x  1]
c.
d.
T2,3 [h( x)  2 x 2  4 x  2]
--------------------------------------------------------------------------------------------------------------------------------------a.
b.
y
y
x
c.
y
x
d.
x
y
x
28