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Precalculus
Pretest- Unit 1
Name:______________________________
Date:_______________________________
MULTIPLE CHOICE (INDICATE YOUR ANSWERS IN THE SPACE PROVIDED):
(1)
Which graph illustrates a quadratic relation whose
domain is all real numbers?
(a)
(2)
(b)
(c)
(1)
(d)
Which graph of a relation is also a function?
(2)
(a)
(3)
(b)
(c)
(d)
Which function is not one to one?
(3)
(a)
(4)
(c)
(d)
If the following graph is y = f(x),
what is the value of f(1)?
(a) -1
(5)
(b)
(b) -2
For which value(s) of x is the function f ( x ) 
(4)
(c) 1
(d) 2
x2  9
undefined?
x 7
(5)
(a) 9
(b) 7
(c) 3
(d) 3 and -3
1
(6)
What is the domain of f ( x )  x  6 ?
(6)
(a) x  6
(7)
(c) x > 6
(d) x = 6
Given the relation {(-2, 3), (a, 4), (1, 9), (0, 7)},
which replacement for a makes this relation a function?
(a) 1
(8)
(b) x  6
(b) -2
(7)
(c) 0
(d) 4
Which of the following is not a linear function?
(8)
(a) x = 8
(9)
(b) 2x + 3y = 6
(c) y = -2x – 1
(d) y = 8
If f(x) = 2x and g(x) = x – 4, what is the value of f(g(3))?
(9)
(a) 2
(b) -6
(c) -2
(d) 6
(10) If f ( x )  3x 2 and g( x )  2x , what is the value of f  g8 ?
(10)
(a) 16
(b) 8 6
(c) 144
(d) 48
(11) A line passes through the points (-1, 4) and (2, 2).
Find the slope of any line perpendicular to this line:
(a) 
3
2
(b) 
2
3
(c)
(11)
3
2
(d)
2
3
(12) Given the set A = {(1, 2), (2, 3), (3, 4), (4, 5)}, if the
inverse of the set is A-1, which statement is true?
(12)
(a) A and A-1 are functions.
(b) A and A-1 are not functions.
(c) A is a function and A-1 is not a function.
(d) A is not a function and A-1 is a function.
(13) What is the inverse of the equation y = 2x – 5?
(13)
(a) y = 2x + 5 (b) y 
1
x  5 
2
(c) y = 5 – 2x
(d) y 
1
x  5 
2
2
Unit 1: Graphs & Functions
Lesson
Homework
Review
Complete pages 4-8 in the packet (2 days)
Difference
HW #4: p. 145: #77 – 80*
Quotient
Symmetry &
HW #5: p. 112: #33, 34, 36, 38 – 45, 48*
Odd/Even/Neither
Functions
Test Review
Problem Set beginning on p
3
Precalculus
Review Material: Functions and Graphs
Name:______________________________
Date:_______________________________
Objective:
To work in groups to rediscover the properties of functions and their graphs from Math 3.
LINEAR FUNCTIONS:
Forms of Linear Equations:
slope-intercept form:
standard form:
point-slope form
PBLM SET
1.
Write the linear equation in slope intercept, standard, and point-slope form given the line passes
through (5, 2) and (7, 9)
2.
Write the equation of the horizontal line that passes through (-9, 2)
Parallel & Perpendicular Lines
Parallel lines have __________ slopes.
Perpendicular lines have slopes that are ____________ ______________.
3.
Write the linear equation in standard form given that the line passes through (-2, 10) and is
4
parallel to the graph of y  3x 
5
4.
Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of
2
4
y x
3
7
4
IDENTIFYING FUNCTIONS
Definitions:
Relation:
Function:
Domain:
Range:
PBLM SET.
1.
State the domain and range of each relation, and state whether the relation is a function or not:
2.
a.
{(-2, 0), (3, 2), (4, 5)}
b.
{(6, -2), (3, 4), (6, -6), (-3, 0)}
Which relation is a function? Why? What is the shortcut rule for determining a function?
(b)
3.
(b)
(c)
(d)
Find the Domain for each:
a.
x2  2x
f ( x) 
x 1
b.
c.
f ( x)  x  3
d. f(x)  - x 2  2x - 27
f ( x) 
1
x 4
2
5
Evaluating Functions: substitute numerical value or variable into function equation and simplify
(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
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Operations with Functions: given functions f and g
sum: f  g( x)  f ( x)  g( x)
product: f  g( x)  f ( x)  g( x)
f
f ( x)
 ( x ) 
, where g( x )  0
g( x )
 g
difference:
quotient:
f  g(x)  f (x)  g(x)
Given functions f and g: (a) perform each of the basic operations, (b) find the domain for each
(1)
f ( x)  5x  4 ; g( x)  x 2  1
(2)
f ( x )  5  x ; g( x )  x  1
Composition of Functions: given functions f and g notation: f  g( x)  f g( x)
Find f  g( x) and g  f ( x) for each f(x) and g(x):
1.
f ( x)  x 2  1; g( x)  3x
2.
f ( x )  x  1; g( x )  x  2
3.
f ( x )  2x  5 ; g( x )  3  x
4.
f ( x)  x 2  x ; g( x)  x  9
5.
f ( x)  2x  3 ; g( x)  x 2  2x
6.
f ( x)  x  1; g( x)  4x 2
7
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**)
Examples: Determine whether the following functions are one-to-one.
1)
2)
3)
f ( x)  x 2  1
f ( x)  2 x  1
f ( x)  3  5 x
Inverse Functions
Steps:
1. Write the equation in terms of x and y.
2. Switch the x with the y.
3. Solve for y.
PBLM SET
1.
Find the inverse of y  4 x  8
3.
2.
Find the inverse of f ( x)  5 x  2
Graph each of the lines in examples one and two and their inverses on the same set of axes and
identify any interesting characteristics.
y
x
4.
1
Given the function f ( x )   x  2
3
(a)
Algebraically, find f -1(x).
(b)
Algebraically, verify your answer to part (a).
8
Precalculus
Lesson- Evaluate functions with variables;
difference quotient
Objectives:


Name:______________________________
Date:_______________________________
evaluate functions as expressions that involve one or more variables
explore functions by evaluating and simplifying a difference quotient
Evaluating & Simplifying a Difference Quotient:
(1)
For f(x) = x2 + 3x + 7, evaluate and simplify:
(a)
(2)
f x  h
(b)
f x  h  f x 
, h0
h
(c)
f ( x )  f (a )
, x a  0
x a
For f(x) = 3x2 – 2, evaluate and simplify:
(a)
f x  h
(b)
f x  h  f x 
, h0
h
9
(3)
(4)
For each given function, find and simplify:
(a)
f(x) = -x2 – 2x – 4
(b)
f(x) = -2x + 5
(c)
f(x) = x2 + 4x + 5
For the function f(x) = 4x – 7, find and simplify:
(a)
(5)
f ( x  h)  f ( x )
, h0
h
f ( x  h)  f ( x )
, h0
h
(b)
f ( x )  f (a )
, x a  0
x a
(b)
f ( x )  f (a )
, x a  0
x a
For the function f(x) = -5x + 2, find and simplify:
(a)
f ( x  h)  f ( x )
, h0
h
10
Precalculus
Lesson- Symmetry, Odd/Even/Neither Functions
Name:______________________________
Date:_______________________________
Objectives:
~To learn an algebraic method for testing for symmetry with respect to axes and origin
~To learn an algebraic method for determining whether functions are even/odd/neither
~To learn shortcut approaches for the above
Odd Functions
symmetric with respect to the origin  TEST: f(-x) = -f(x)
Even Functions
symmetric with respect to the y-axis  TEST: f(x) = f(-x)
Symmetry Tests
symmetric with respect to the:
y-axis
x-axis
origin
the given equation is equivalent when:
x is replaced with -x
y is replaced with –y
x and y are replaced with –x and -y
Examples:
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Unit 1: Graphs & Functions
Definitions, Properties & Formulas
Linear Equation
Slope
equation of a straight line
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
Parallel 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 nonvertical lines in a plane are parallel if and only if their slopes are equal
12
and they have no points in common. (Two vertical lines are always parallel.)
ex)
Perpendicular
Lines
y = 2x + 3
m=2
and
y = 2x – 4
m=2
 equal slopes
 // lines
Two nonvertical lines in a plane are perpendicular if and only if their slopes are
negative reciprocals. (A horizontal and a vertical line are always perpendicular.)
5
2
y  x 7
y   x 1
ex)
and
 neg. recip. slopes
2
5
5
2
  lines
m=
m= 
2
5
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
a relation in which each element of the domain is paired with exactly one element
in the range.
Vertical Line
Test (VLT)
Horizontal Line
Test (HLT)
One-to-One
Functions
Inverse Relations
& Functions
Writing Inverse
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)
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)
Odd Functions
symmetric with respect to the origin  TEST: f(-x) = -f(x)
Even Functions
symmetric with respect to the y-axis  TEST: f(x) = f(-x)
Symmetry Tests
Operations with
Functions
symmetric with respect to the:
y-axis
x-axis
origin
the given equation is equivalent when:
x is replaced with -x
y is replaced with –y
x and y are replaced with –x and -y
sum:
(f + g)(x) = f(x) + g(x)
difference:
(f – g)(x) = f(x) – g(x)
13
Composition of
Functions
product:
(f  g)(x) = f(x)  g(x)
quotient:
f
f ( x)
 ( x ) 
, where g( x )  0
g( x )
 g
given functions f and g, the composite function is f  g( x)  f g( x) , where g(x)
is substituted for x in the f(x) function
Precalculus
Review- Unit 1 Test
Name:______________________________
Date:_______________________________
SHOW ALL WORK!
(1) Which of the following equations define functions? Explain your reasoning.
(a) y = x + 6
(b) y2 = x + 1
(c) y3 = x + 4
(d) y = x – 2
(e) y3 = x – 3
(f) y2 = x – 5
(2) Which of the following functions are one-to-one? Explain your reasoning.
(a) f(x) = x5
(b) g(x) = 2x + 7
(c) h(x) = x2 – 4
(3) Write the linear equation in standard form Ax + By = C that passes through the points (3, 5) and
(4, -3).
14
(4) Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3)
and is parallel to the line 2x + 2y = 5.
(5) Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3)
and is perpendicular to the line 3x – 3y = 4.
(6) Write the linear equation in standard form Ax + By = C that is a horizontal line and passes
through the point (-9, 2).
(7) Given f(x) = 2x2 – x + 3 find f(k – 2)
15
(8) Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1)
16
(9) For the function f(x) = -6x + 5, find and simplify:
f ( x  h)  f ( x )
, h0
(a)
h
(10) Find the domain of each of the following:
2x  3
(a) f ( x ) 
2x
(b)
f ( x )  f (a )
, x a  0
x a
(b) g( x )  x  4
(11) Algebraically determine the symmetry with respect to the y-axis, x-axis, and origin, if any
exists, for each of the given equations:
(a) 2x – 4y = 7
(b) 9x2 – 4y2 = 36
(12) Algebraically determine if each of the given functions are odd, even, or neither:
(a) f(x) = x2 – 6x
(b) f(x) = x6 + 7
17
18
(13) Perform the four basic operations with the functions f(x) = 4x and g(x) = x2 + 2
(f + g)(x) =
(f – g)(x) =
(f  g)(x) =
f
 ( x ) 
 g
(14) Given f(x) = x2 – 3 and g( x )  2x  4 , find f  g( x) and g f ( x) and give each domain.
(15) For each of the following functions, f(x), find the inverse, f -1(x):
(a) f(x) = 5x + 2
4
(b) f ( x ) 
x3
(16) Given the graph on the right, answer the following questions:
(a)
write the linear equation in slope-intercept form:
(b)
write in slope-intercept form the equation of the parallel line
through (2, -1) and graph:
(c)
y
x
write in slope-intercept form the equation of the perpendicular
line through the x-intercept and graph:
19