Functions and Relations
Use the first row
1. Each student in this room is asked to select their favorite problem on the board from last night’s HW.
2. Each student is asked which of the first five problems they believe to be correct
3. Each student is asked to select a student from the second row.
We do the same idea as above but with numbers
Relation: a correspondence between two sets –
We can write as ordered pairs of the form (x,y), use a graph, or we can use an equation to represent the relation
Sets
Graphs
Equations
Function: a relation in which each x has only one value of y associated with it.
Domain: set of all permissible value of x
Range: set of all permissible values of y ( obtained by using permissible values of x)
sets:
a)
b)
1
2
3
4
4
Relation, Function, both, neither ?
Domain:
_____________________
Range:
1
2
1
-1
-2
Relation, Function, both, neither ?
Domain:
_____________________________
Range:
_____________________
_____________________________
c)
d)
2
5
10
17
0
1
2
3
4
1
2
4
8
0
9
7
4
Relation, Function, both, neither ?
Domain:
_____________________
Relation, Function, both, neither ?
Domain:
_____________________________
Range:
Range:
_____________________
_____________________________
Graphs:
Vertical line test:
construct vertical line – if each line crosses graph only once, then we have a function
if more than one crossing point, then it is just a relation
y=4x
y=|x|
a)
c) y2 = x2 + 1
b)
d) y2 = x+1
equations
a) y = 2x – 3
b) x2 + y2 = 4
c) y = log b x
d) xy = 3
Graph each of the following.
a) 2x = y – 4
b) x2 = y – 2x + 1
c) f(x) = 3x
d)
g(x) = log4 x
Which of these relations are also functions ?
a)
2x
y = ----------------x2 + 5x - 14
b)
| y | = 2x
c) x2 + 2x + y2 = 4
d)
y + 4= 0
Find the domain of
a)
y = 2x – 1
c) y =
b) y =
x 2  2x  3
x2
d) y =
4. Find the range of
a) y = x2 + 2
c) circle of radius 2 with center at (2, -1 ).
x2
x3
b) y = x + 2
x2
Problem Set #16
Linear Equations
An equation of the form ax + by = c is called a linear equation
ex. 2x – 3y = 4,
4x = 6x + 1 ,
ex. 0x + 2y = 4
x = - 2y + 4,
x = - 3,
y= 4
ex. 3x – y = 6
ex. 3x + 0y = 9
There are three types of lines.
horizontal, vertical, and slant lines.
A horizontal line has slope zero and because it crosses the y –axis, its equation is of the form y = b
A vertical line has an undefined slope and because it crosses the x-axis its equation is of the form x = a
Slant lines have a slope and are of the type y = mx + b
We can find the slope by using
m=
y 2  y1
rise y

=
or by writing an equation in the form y = mx + b
run x
x2  x1
ex. Find the slope of
a) y = 3  _________
c) 2x – y = 3  ______
b) x = - 3  _____
ex. Find the equation of the line that
a) is horizontal and passes through ( 4, 7 )
b) is vertical and passes through the point ( -1, 5).
c) passes through (-1, 4) and has slope 2.
d) is parallel to 2x + 3y = 1 and passes through ( -1, 4)
e) passes through ( 4, -1) and ( 3, 0 )
f) is perpendicular to 2x – 3y = 4 and passes through the point (2, -3 )
Other Material: Use of Quadratic Equations
x2 + 12x – 64 = 0
Find the sum of the roots ( solutions ) . ______________
product of the roots . _____________
What about
x2 – 2x - 123 = 0
sum = _____________
product = _____________
Now try,
21x2 + 4x - 32 = 0 ==> sum = _____________
product = ______________
Notation:
if f(x) = x2 + 2x – 1,
then f(0) = ______________
f(2) = _____________ and f(h) = ___________
Problem set #17
Quadratic Functions
Quadratic Functions: The graph of a function of the form
f(x) = ax2 + bx + c or y = ax2 + bx + c is
a parabola that opens up if a > 0 , opens downward if a < 0
with vertex V ( - b/2a, f(-b/2a) )
We can find the x-intercept, the y-intercept, and a couple of points to get an idea of the graph of the function.
ex. Sketch the graph of f(x) = - 2x2 – 4x + 1
ex. Sketch the graph of g(x) = 4x2 + 2x
ex. What is the maximum value of h(x) = - 2x2 + 3 ? and where does it occur ?
ex.
The sum of two numbers is 28. Find the two numbers so that their product is a maximum.
ex. find the minimum ( maximum) :
f(x) = ½ x2 + 4x
ex. profit: P = 16x - 0.1x2 - 100
a) at what level of production is the profit at its maximum ?
b) What is the maximum profit ?
ex. I have 150 feet of fencing. what should the dimensions of my rectangular yard be if the area enclosed is as large as
possible.
ex. 42/290: f(x) = 104.5x2 - 1501.5x + 6016 → models the death rate per year per 100,000 males, f(x) , for US men who
average x hours of sleep each night. How many hours of sleep, to the nearest tenth of an hour, corresponds to
the minimum death rate ? What is this minimum death rate, to the nearest whole number ?
ex. 57/291: f(x) = - 0.018x2 + 1.93x - 25.34 describes the miles per gallon, f(x), of a Ford Taurus
driven at x miles per hour. Suppose that you own a Ford Taurus. describe how you can use
this function to save money.
Functional Notation:
Let f(x) = x2
and g(x) = 2x + 4
Find
a) f + g:
b) fg :
c) f/g :
d) composition of functions --f o g : ( f (g(x) ) ) =
2. Find each of the four values above for x = 1
a) (f + g )(1) = __________
b) (fg)(1) = _________________
c) ( f/g) ( 1 ) = __________
d) ( f o g ) ( 1 ) = ______________
e) f ( h ) = ______________
f)
g( 2 + h) = __________________
Problem set #18
Logarithms and Exponential Functions
Exponential:
We write y = ax or f(x) = ax
ex. let f(x) = 2x ,
find f(0) = ________, f(1) = _______, f( 2) = ________ f( -1 ) = ________
f( -2) = ______
x
f(x)
0
1
2
-1
-2
-3
We get a graph for this function - is it really a function ? _____
This idea would work with any exponential function of the form f(x) = a x.
What is the graph of y = 2x ?
x
0
f(x)
1
2
-1
-2
-3
What about y = - 2x
x
f(x)
0
1
2
-1
-2
-3
Exponential Functions:
General equation: f(x) = ax
Graph
Some examples: y = 12 x,
g(x) = 2 x + 3
Graphs of
y = a-x
x-intercept: ____________
y = - ax
y = - a –x
y-intercept = ____________
What about equations like y = 3 + 2x, what is the y-intercept ? the x-intercept ?
49/383
52/383
55/383
Also, find ( 1 + 1/m ) m as m gets larger and larger ( as m → ∞ ) . ( 1 + 1/m)m → _______
Find the domain, the range, x and y –intercepts of
Logarithms
Logarithm: We write y = logb x or f(x) = logb x
We say “the logarithm of x base b” to mean there is an exponent y so that b y = x.
ex. Log 5 125 = y → 5y = 125 → y = ? _________
log 64 8 = y → 64 y = 8 → y = ? _________
ex. let f(x) = log 2 x
find
f( 0 ) = __________ f(1) = _________ f(2) = ______________
f( 4) = ________
f( ½ ) = __________
x
f(x)
0
1
This is the general graph for y = log b x
Examples:
. Find x so that 128 = 2 x , x = _________
What about 345 = 2x, x = _____________
f( 1/8 ) = _______ What about f( - 2) = ? _________
2
½
¼
-1
Properties of Exponents and Logarithms:
1) Domain
of y = ax ==>
2) Range
of y = ax ==>
3) x-intercept
of y = ax
Other Properties of Exponents and Logarithms.
1. log b xy = log bx + log b y
2. log b (x/y) = log b x - log b y
3. log b (xk ) = k log b x
Other properties
4. log b 1 = 0
5. log b b = 1
6. log b 0 = undefined
7. log b ( x) = undefined if x < 0
of y = log b x ==>
of y = log b x ==>
of y = log b x
IF b = 10 ,
we write log 10 x = log x and call it the common logarithm
If b = e ( where is the irrational number e ),
we write log e x = ln x ---- and call it the natural logarithm
Domain: Find the domain of y = log b ( x + 2 )
Find the domain of y = ( x2 – 2x – 3 )
examples:
81/396
84/396
Note: log x2 = 2 log x so do they represent the same thing ?
y = log x2
and
In other words look at their graphs
y = 2 log x
ex. Find the solution of
log2 x -
log2 (x - 2 ) = 1
log 2 x + log 2 (x – 3 ) = 2
what about
log2 x -
log2 (x + 2 ) = 1 ?
Other examples
ex. Find the domain of
a) y = log 3 ( 2x – 1 ) → __________________________________________
2 x  3 → ___________________________________________
b) f(x) =
c)
g(x) = log 2 ( x 2 - 2x – 8 ) → ____________________________________
d) h(x) =
x 2  2 x → __________________________________________
ex. Find x if
a) 2
log2 7
 x → x = _______________________
b) log2 x + log2 (x+1 ) = 1 → x = ________________________
c) log4 165 = x → __________________
d) log x - log (2x – 1 ) = 0 → ________________
e)
If log b 16 = 0.21, then find
log b 2 = __________
Problem set #19
Chapter 6. Solving Polynomial Equations
Long hand division –
ex. 12  5 = ___________
ex. Suppose you had 17 apples that were to be evenly divided by five individuals. How much should
each one get so that nothing remains ?
ex. Find ( x2 - 4 )  ( x + 2 ) = _________________
ex. Find ( x2 + 3x - 4 )  ( x – 1 ) = _______________
ex. Find ( x2 + 2 )  ( x + 2 ) = ______________________
The remainder Thm.
Let P(x) be a polynomial with real coefficients. The remainder of P(x)  ( x – r ) is the same as P(r).
ex. Find the remainder of
(x2 + 3x - 4 )  ( x – 1 )  _______________
( x2 + 2 )  ( x + 2 )  __________________
What happens when the remainder is zero ? ___________________________________
The Factor Thm and its converse.
If (x – r) is a factor of the polynomial P(x), then r is a root of P(x) = 0
ex. x2 – 4x – 5:
we can see that x – 5 is a factor and what are the solutions of x2 – 4x – 5 = 0 ? _________
another factor ? _________
If r is a root (zero, solution of ) of P(x) = 0 , then x – r is a factor of the polynomial P(x) = 0
ex. when we solve the equation x2 – 4x = 0 we get x = ____________
find the factors.
______________
Use of the Remainder and Factor Theorems.
1) Is ( x + 1 ) a factor of ( x4 - 5x - 4 ) ? _____________________
2) Is ( x – 2 ) a factor of 3x3 - 9x – 6 ? ____________________________
3) is x = 3 a solution of the equation x3 – 6x – 9 = 0 ? Can you find all of the factors of x3 – 6x – 9 ?
4) Factor x3 + 2x + 1 by using the fact that x = -1 is a solution of the equation x3 + 2x + 1 = 0
Sometimes finding the remainder is not sufficient. Finding the quotient may be useful and in that case
the remainder thm. is not sufficient. We can use long-hand division or Synthetic Division.
Synthetic Division
shorthand way of dividing two polynomials where the divisor is of the form x – r.
ex. (x2 + 2 )  ( x + 2 ) = ____________
ex. Find ( x4 - 5x - 4 )  ( x + 1 ) = __________
ex.
Find all of the roots ( solutions ) of x3 + 2x + 1 = 0
ex.
Find all zeros of the polynomial P(x) = x3 + 1.
Problem set #20
A polynomial P(x) can always be written in the form a nxn + an-1xn-1 + … + a2x2 + a1x + a0
example: 3x4 + x2 – 2x + 7  __________________
4x3 + 2x – 3  _________________
The degree of a polynomial provides information as to the number of roots (solutions) the polynomial
equation will have. We can use the factor thm to arrive at the following conclusion.
Fundamental Theorem of Algebra
Let P(x) be a polynomial with real coefficients and of degree n. Then P(x) has n roots which
1) may or may not be distinctive
2) may or may not be real
ex. x2 + 4 = 0 has how many roots ? __________ and they are both ? __________
ex. 4x2 - 9 = 0
ex.
x2 - 4x + 4 = 0
____________
_____________
________________
__________________
Now we find all of the roots of the equation x3 + 1 = 0 . there are _______ roots and they are
____________
Descartes’ Rule of signs:
can be used to reduce the number of possibilities(roots).
If the original polynomial P(x) has no sign variations, then it has no positive roots
If P( - x ) has no sign variations , then P(x) has no negative roots.
By itself Descartes’ Rule of signs is not very helpful but when used with the following thm. , it is useful in finding roots of
some polynomial equations.
ex. x4 + 3x2 + 2 = 0  Find all of the roots. How many of them are positive ? ___________
How many are negative ? ______________
ex. What about x5 - 2x - 3 = 0 → positive ? ______________
negative ? __________
Conjugate Pairs Thm.
Let P(x) be a polynomial with real coefficients. If a + bi
is
____________.
ex. x2 + 9 = 0  _______________
ex. x4 + 5x2 + 4 = 0  _____________________
is a solution (root) of P(x) = 0, then so
Quadratic Pairs:
Let P(x) be a polynomial with rational coefficients. If
perfect square, then so is
_______________
ex. x2 + 6x - 5 = 0
a b
is a solution of P(x) = 0 , b not a
ex.
x2 - 3 = 0
Rational roots:
Let P(x) be a polynomial with rational coefficients. If r is a rational solution of the equation
P(x) = 0,
then r can be written in the form r = p/q, where p is a factor of the constant term and q is a factor
of the leading coefficient of P(x).
ex. x3 - 4x + 3 = 0
ex. 2x4 + 3x2 - 5 = 0
c=3
and leading coefficient is 1
Additional Examples
Synthetic Division the remainder thm is useful but it does not provide a quotient.
(x2 + 2x + 1 )  ( x – 1 ) = ___________________
(x3 + x + 2 )  ( x + 2 ) = _________________
Find P( 4 ) if P(x) = 2x3 – 2x + 1 _________________
Is 4 a solution of P(x) = 0 ? Why or why not ?
Is x – 4 a factor of P(x) ? Why or why not ?
Find ( 2x3 - 2x + 1 )  ( x – 4 ) = ____________________
Is x + y factor of x3 + 2xy2 – y3 ?
Find all of the roots of x3 - 2x + 1 = 0 if x = 1 is known to be a solution.
Find all of the zeros of the polynomial x3 - 2x2 + x – 2 = 0
Find all of the zeros of x4 - 2x3 + 5x2 - 8x + 4 = 0
Problem set 21
Binomial Expansion
( a + b)n = _______________
ex. ( a + b)0 = _____________
ex. ( 2x + y)1 = ____________
ex. ( x – y )2 = ______________________________
ex ( x + 2y )3 = _______________________________
ex. ( x + y)5 = _________________________________
In general we find patterns that allow us to find specific terms of an expansion without having to find all of the terms.
ex. Find the first two terms of the expansion of
( 2x – 1/x )6 = ____________________________
ex. Find the last two terms of the expansion of ( x + 1/x) 7 = __________________________
ex. How many terms are in the expansion of ( 3x + 2y ) 12 ? _______________________
ex. We can find any term along the way, say the 7 th term of ( x2 + x)12 → ______________________
Inequalities in two variables ( on the plane )
Find the solution of the following inequalities.
x + 2y > 4
2x – y < 2
x<4
Find the solution of the following system of inequalities.
x + 2y > 4
2x – y < 2
2x – y < 2
x<4
Problem set #22
System of Equations
Substitution :
2x – 3y = 6
x + 5y = 3
1) decide which variable in what equation to solve for:
2) Solve for that variable in that equation:
3) Substitute in the other equation:
4) Go back and use equation from 2 to obtain the remaining part of your solution:
→ solution: (x,y) =
Another example:
3x – 6y = 3
2x + 4y = 2
Elimination:
x + 4y = -2
3x – 2y = 8
1) decide which variable to eliminate:
2) Get the LCM of the coefficients of the chosen variable:
3) ( Add-subtract) to eliminate variable and create a new equation without variable.
4) Solve for remaining variable.
5) Back substitute into original equations ( any one of them ) to solve for remaining variable .
Solution: (x, y )
Another example:
3x – 12y = 1
2x - 8y = 3
Additional examples:
page 452:
1)
5)
17)
19)
29)
43)
47)
System in three variables:
Reduce to a system in two variable by eliminating one variable and creating two new equations in
only two variables. Solve the new system of two equations and two variables.
x + 2y + 3z = 7
2x – y – 4z = -1
x + 2y – z = 5
1) eliminate ____
a) use equations: ___ and __
b) use equations: ___ and _____
2) Solve the new system
3) Final Solution:
Another example:
x + 2y – z = 5
x+ y
=3
x z = 2
More Examples on page 481
2)
8)
14)
20)
Problem set #23
Variation: direct and inverse
We say that y varies directly as x if
there exists a constant k so that y = kx.
We say that y varies inversely as x if
there exists a constant k so that y = k/x.
ex. 32/364
ex. 34/ 364
ex. 38/364
ex.
39/364
ex. 45/ 364
Matrices
General Notation:
1) rectangular array of numbers with rows and columns
We normally use capital letters to name the matrices.
A = 3 ,
B = 3 1 / 5,
2
3
4
5
1
6
7
8
9
0 

D=
, E=
  1 10  2  3  4


 5  6  7  8  9 
3  2
C= 
,
5 7 
4
7 
 
2) Dimension of a matrix: m x n
We use the number of rows and columns to describe the matrix.
A is a ___________ matrix
C: __________
B is a ____________ matrix
D: _____________
E: ____________
3) elements of a matrix: aij
Look at matrix C: we can label the elements of C as follows:
Look at matrix E: we can label the elements of E as follows:
Look at matrix D: find each of the following entries (elements)
d13 = _________
d25 =__________
d32 = _________
d42 = __________
Special Types of Matrices:
Zero Matrices:
All entries are zero
2x2 zero matrix
1x5 zero matrix
4x3 zero matrix
Square Matrix: A matrix that has the same number of rows as columns
1 0 0
1 2
A = 5 , B = 
C = 0 1 0 



3 4
0 0 1
Identity Matrices: diagonal entries – a11, a22, a33,.... are all = 1 while all other entries = 0
1 ,
1 0
0 1  ,


1 0 0
0 1 0  ,


0 0 1
1
0

0

0
0 0 0
1 0 0
, ....
0 1 0

0 0 1
Addition: add corresponding entries so that you end up with a matrix that resembles the original
two in size- this can
only occur if the original matrices are identical in size .
A + B is defined if A : m x n matrix, then B must also be m x n matrix.
6
3  2 +   = _________
 2
1  2 4  3  ___
2  1 + 0  3 =  ___

 
 
0 4  1 2   ___
___ 
___ 
___ 
Subtraction: if treat matrices as real numbers, we can use addition.
Let - A represent the opposite of matrix A. Then B – A = B + ( -A).
2  3  3  2
 ___
4 0  -  1 1  =  ___

 


 4   2 
 ___ 
  2 -   2 =  ___ 
   


 0   3 
 ___ 
___ 
___ 
There are two types of products of matrices –
multiplication by a scalar (nonmatrix – real number)
multiplication of two matrices
Scalar Multiplication: easy product - distributive law
 3   ___ 
a) 4   = 

  2  ___ 
 2 1  ___
c) - 2 
 = 
 2 1  ___
b) - 2 2  3 1 0 = ___ ___ `___
___ 
___ 
___ 
Some Simple products of Two matrices:
If we multiply matrix A by B( in that order), then the number of columns of A must be the same
as the number of rows of A. If A is an m x p matrix, then B must be a p x n matrix
ex.
2
 1 • 1  2 = ?
ex.
4
1  2 3 •  0  =
 3
 1  2  1
  2  • 1 0  = ?
  

ex.
4
1 • 2  3 =
 
In the two examples above, what do you get if you change the order of the matrices ?
 1 2 
ex. 1 2 3 •  0  2 =
 4
1 
General Product of Matrices
1 2
 1  2 3
ex. 
•

  2 2 4 =
3 4


2  3
 1  2 3
ex. 
• 2 3  =



  2 2 4
4 1 
ex.
Sequences
Factorials:
Def. n! = n(n-1)(n-2)    (2) (1)
ex. 4 ! = 4(3)(2)(1) = 24
ex. 6 ! = ______________
ex. 100 ! = ______________
We define 1 ! = 1 and 0! = ______
Find 5 ! = ________
4 ! / ( 5 ! - 7 ! ) = ____________
240! / 241 ! = ______
Sequences:
a1, a2, a3, …
a correspondence between the set of natural numbers and a second set ( we can list the numbers in a list, 1st, 2nd, 3rd, … )
We can have a finite sequence; there is a beginning term and an ending term
a1, a2, a3,… an  here an represents the last term and n represents the number of terms in the
sequence.
We can have an infinite sequence;
a1, a2, a3, …, an,…  here an represents a general term of the sequence, the 3rd , the 10th, …
1, 4, 7, _____, ______
12, 5, - 2, ________, ________
1, 3, 4, 7, 11, ______, _________
2, 6, 10, 18, 34, ________, _________
2, - 4, 8, _________, ___________
16, 4, 1, ________, __________ , ________
-2, 0, 2, 0, -2, 0, 2, __________, _________
1, ½, 1/3, ________, ________, _________
2, ½, 3, 1/3, ______, _________,
½, 2/3, ¾, 4/5, ______, ________
x, 3x – 1, 5x – 2, ....
2, x + 4, x2 + 6x + 8, ....
There are several ways to describe a sequence.
By its position ( the value of n). If an represents the fifth term, then n = 5, its position.
ex. if an = 3n + 1
ex. an = ( -1)n - 1 ,
then a1 = ______, a3 = __________
a1 = ________, a2 = __________
a25 = _________
a3 = ________, a20 = __________
By using preceding terms in the sequence, an represents the current term in question,
while an-1 represents the preceding term, an-2 represents the term right before the preceding term,…
ex. an = ( an-1 ) 2 ,
a1 = - 2,
ex. an = 2 - an-1,
a1 = 3,
a2 = __________, a3 = __________ , a20 = ________
a2 = _________ a3 = ___________,
Summation of a sequence:
Suppose you wanted to find the sum the first five terms of the sequence defined by
an = 2n, we can easily list the five terms and find their sum.
We can also write
 an or
Find each of the following sums:
1)
2)
3)
 2n to represent the sum.
a5 = __________
Three types of sequences and progressions.
Arithmetic Progressions (AP ): need 1 st term (a1) and common difference ( d )
Geometric Progressions (GP ) : need 1 st term (a1 ) and the common ratio ( r )
Harmonic Progressions (HP):