1 Precalculus Review 0.1 The Real Numbers and the Cartesian Plane 0.2 Functions 0.3 Geometric...

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Precalculus Review

0.1 The Real Numbers and the Cartesian Plane

0.2 Functions

0.3 Geometric Properties of Functions

0.4 Linear Functions

0.5 Quadratic Functions

0.6 Polynomials, Rational Functions, and

Power Functions

0.7 Applying Technology

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The Real Numbers and the Cartesian Plane

The real number include the familiar integers, such as -1, 0, 1, 2, etc., and fractions,

such as2

3,

1

10

,

21

5. The collection of integers and fractions is called the set of

rational numbers.

However, the rational numbers do not make up the entire real number system. There

are also irrational numbers, such as 2 , and .

The above numbers are shown in Figure 0. 1. 1.

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Intervals:

( , )a b stands for the set of x satisfying :a x b

[ , ]a b stands for the set of x satisfying :a x b

Absolute value: Definition 0. 1. 1

The absolute value of a real number x , denoted by x , is defined as follows:

f 0{

if 0.

x i xx

x x

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The Cartesian plane To set up the correspondence, we take two axes that meet perpendicularly at their

origins. We usually take one axis to be horizontal with the positive direction to the

right and call it the x axis . The other axis, called the y axis , is taken to be vertical

with the upward direction as positive.

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The point P at which we arrive corresponds to the pair ( , )x y . The first number of the

pair is called the x-coordinate of P, and the second is called its y-coordinate. A few

examples are shown in Figure 0. 1. 2.

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Distance in the plane and the equation of a circle By using the Pythagorean theorem (see Figure 0. 1. 3),

we see that the distance d from the origin (0,0) to a

point ( , )P x y of the plane is given by the formula

2 2d x y .

More generally, the distance between two

points 1 1( , )A x y and 2 2( , )B x y is given by the

formula

2 22 1 2 1( ) ( )d x x y y .

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Equation of a circle For any positive number r , the

equation 2 2 2x y r represents a circle of

radius r centered at (0,0) (see Figure 0. 1. 4).

.

More generally, the equation of the circle with

center at a point 0 0( , )x y and radius r is given by

the formula

2 2 20 0( ) ( )x x y y r

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Functions

Definition 0. 2. 1 A function f is a rule that assigns a unique value ( )f x to every number x in a given

set D .

Example 0. 2. 1 Let f be the function defined by the formula

2( ) 3 15 18f x x x

with the set of all real number as domain. Find the values obtained by setting

(i) 1, (ii) 2, (iii) 1.x x x t

SOLUTION 2(i) ( 1) 3( 1) 15( 1) 18 36.f

2(ii) (2) 3(2) 15(2) 18 0.f 2 2(iii) ( 1) 3( 1) 15( 1) 18 3 9 6.f t t t t t

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Functions

When is a curve the graph of a function? It is important to note that not every curve in the plane is the graph of a function. A

curve may be given by an equation in x and y , but it is not the graph of a function

unless we can put the equation in the form ( ).y f x

The Vertical Line Test

A curve in the plane is the graph of a function of the form ( )y f x if and only if no

vertical line meets the curve more than once.

The Horizontal Line Test

A function ( )f x is one-to-one if and only if no horizontal line meets its graph more

than once.

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Functions

Inverse functions In general, if two function f and g are related in that way-that is, if

( ) ( )y f x x g y ,

we say that they are inverses or that each is the inverse of the other.

One other point worth noting about inverse functions, f and g , is

this:

the domain of f is the range of g , and

the domain of g is the range of f .

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Geometric Properties of Functions

Increasing and decreasing functions We say that ( )f x is increasing if its graph rises to the right

Definition 0. 3. 1 A function ( )f x is increasing on an interval ( , )a b if

1 2 1 2( ) ( ) wheneverf x f x a x x b .

In other words, the value of ( )f x becomes greater as x increases over the

interval ( , )a b .

Definition 0. 3. 1 (0. 3. 2) A function ( )f x is increasing (decreasing) on an interval ( , )a b if

1 2 1 2 1 2( ) ( ) ( ( ) ( )) wheneverf x f x f x f x a x x b .

In other words, the value of ( )f x becomes greater (smaller) as x increases over the

interval ( , )a b .

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Symmetries in graphs In general, the graph of a function ( )f x is

symmetric about the y axis if the

function satisfies the condition:

( ) ( )f x f x ,

the function is also called even function

as shown in Figure 0. 3. 4.

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Symmetries in graphs Another important type of symmetry is

shown in Figure 0. 3. 5, we say that a

graph with that property is symmetric

about the origin. This kind of function

satisfies the condition:

( ) ( )f x f x

and is called an odd function.

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Linear Functions

Definition 0. 4. 1 A linear function is a function of the form ( )f x mx b , where m and b are given

numbers. We refer to m and b as constants.

The graph of the function f(x) is shown in Figure o. 4. 1.

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Linear FunctionsThe slope The difference in height between the first and second points is 2 1y y , sometimes

referred to as the rise. The horizontal change from one point to the other is 2 1x x ,

sometimes called the run. The quotient of these two differences-the rise over the

run-is called a difference quotient.

The triangles shown in Figure 0. 4. 2 are similar, so that

4 3 2 1

4 3 2 1

y y y y

x x x x

.

We sometimes write the difference quotient in the form

y

x

The difference quotient is called the slope of the line, usually denoted by m .

To summarize:

2 1

2 1

Slopey y y

mx x x

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Linear Functions

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Linear Functions

Example 0. 4. 1 Find the slope of the line connecting the

points ( 1, 2) and (2,3) .

SOLUTION:

Putting these numbers into formula

2 1

2 1

Slopey y y

mx x x

gives

3 ( 2) 5Slope .

2 ( 1) 3

The line is shown in Figure 0. 4. 3.

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Linear Functions

The slope measures the steepness of the line- how much the graph rises (or fall)

vertically for each unit of change in the horizontal direction.

Figure 0. 4. 4 shows the four possible.

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Linear Functions

Definition 0. 4. 2 Given a set of data points-say, 1 1( , )x y , 2 2( , )x y , and so forth, up to ( , )n nx y , where n is

some positive integer-the regression line is given by the equation y mx b , where

1 1 1 12 2 21 1

( ... ) ( ... )( ... )

( ... ) ( ... )n n n n

n n

n x y x y x x y ym

n x x x x

and

1 1( ... ) ( ... ).n ny y m x x

bn

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Quadratic Functions

Definition 0.5. 1 A quadratic function is a function of the form

2( ) ,f x ax bx c

where a , b , and c are constants with 0.a

To find the roots of such a function, that is, to solve the equation 2 0,ax bx c

we have the quadratic formula: 2 4

.2

b b acx

a

The quantity 2 4b ac is called the discriminant.

There are

two distinct solutions if 2 4 0,b ac

a single solution if 2 4 0,b ac and

no solutions if 2 4 0.b ac

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Quadratic Functions

Factoring a quadratic and determining its sign If a quadratic function, 2( ) ,f x ax bx c has the real numbers r as its

roots-if ( ) 0f r and ( ) 0f s , them we can have the following:

2 ( )( ).ax bx c a x r x s

If 1a , the above formula reduces to

2 2( )( ) ( ) .x bx c x r x s x r s x rs

So that ( )b r s and .c rs If 1a , you can write the quadratic as

2 2( ).b c

ax bx c a x xa a

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Quadratic Functions

Example 0. 5. 1 Factor each of the following quadratics:

2 2 2(i) 5 6, (ii) 4 8 3, (iii) 1.x x x x x x

SOLUTION

(i) We look for numbers r and s , 2 5 6 ( 2)( 3).x x x x

(ii) Writing 2 2 34 8 3 4( 2 )

4x x x x , then 2 1 3

4 8 3 4( )( ).2 2

x x x x

(iii) In this case we need 1r s and 1rs , we fall back on the quadratic formula,

which gives the roots:1 5

2

.

Thus, we obtain the factorization: 2 1 5 1 51 ( )( ).

2 2x x x x

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Quadratic Functions

The graph of a quadratic function The graph of a quadratic function is called

a parabola. In Figure 0. 5. 3 we see six

examples. The equations corresponding to

these six graphs are as follows:

2 2 21, , 2 ,

2y x y x y x

2 2 21, , 2 .

2y x y x y x

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Quadratic Functions

These functions are all of the form 2y ax . Moreover, the graph of any such function:

opens upward if 0a , with (0,0) the lowest point of the graph and

opens downward if 0a , with (0,0) the highest point of the graph.

For instance, the graph of 22 5y x , shown in Figure 0. 5. 4, is obtained from

that 22y x by a shift of five units upward.

If we replace x by ( 2)x , we get 22( 2) 5y x , whose graph is shown in Figure 0.

5. 5.

In general, by completing the square, we get 2

2 2 4( ) , with and .

2 4

b ac bax bx c a x h k h k

a a

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Quadratic Functions

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Quadratic Functions

Optimization of quadratic functions If 2( )f x ax bx c , then by completing the square, we see that:

1. ( )f x has a minimum value if 0a and a maximum value if 0a ;

2. the minimum or maximum occurs at / 2x b a ;

3. the value of ( )f x at / 2x b a is 2(4 ) / 4ac b a .

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Polynomials, Rational Functions, and Power Functions

Definition 0. 6. 1 A polynomial of degree n is a function of the form

22 1 0( ) ... ,n

nf x a x a x a x a

where 0 1, ,... na a a are constants, and 0na .

The number na is called the leading coefficient.

Figures 0. 6. 3 and 0. 6. 4 show the graphs of two fourth-degree polynomials.

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How does the graph behave as it moves far toward the left?

Given 22 1 0( ) ... ,n

nf x a x a x a x a

if n is even and 0na , the graph rises to the right and left;

if n is even and 0na , the graph falls to the right and left;

if n is odd and 0na , the graph rises to the right and falls to the left;

if n is odd and 0na , the graph falls to the right and rises to the left.

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A simple way to remember how the graph behaves as x goes further and further to the

right (or left) is the following principle: A polynomial behaves more and more like its

highest-degree term as the absolute value of x gets larger and larger.

If x is very large in absolute value, 22 1 0...n n

n na x a x a x a a x .

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Polynomials, Rational Functions, and Power FunctionsThe graphs of ny x for n from 1 to 6 are drawn in Figure 0. 6. 5, all on the same set

of axes. Figure 0. 6. 6 shows the graphs of ny x for n from 1 to 6.

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Polynomials, Rational Functions, and Power FunctionsDefinition 0. 6. 2 A rational function is a quotient of two polynomials.

When a graph approaches a vertical line in the way that the graph

of 1/y x approaches the y-axis-either climbing or falling unboundedly- we call that

line a vertical asymptote of the graph.

And when a graph approaches a horizontal line as it moves farther and farther to the

right or left as the graph of 1/y x approaches the x-axis-we call that line a

horizontal asymptote of the graph.

In general, a rational function has a vertical asymptote at each point where its

denominator is zero.

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Polynomials, Rational Functions, and Power FunctionsPower functions A power function has the form

my Cx ,

where C and m are fixed numbers.

Notice that

if m is a positive integer, it is a polynomial, and

if m is a negative integer, it is a rational function.

Suppose m is a rational number, say /m p q , where p and q are positive integers.

In that case, we define

/ qp q px x .

If m is negative, say, /m p q for positive integers p and q , then we use the formula

//

1p qp q

xx

.

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Applying Technology

Computers and graphing calculator can be of great help in analyzing functions

Graphing with a calculator To graph a function given by a formula, choose certain options, and press a key to

display the graph. Figure 0. 7. 1 shows a typical calculator screen.

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Applying Technology

To create an Excel chart from a table of data, we select Chart from the Insert menu

listed in the heading of Figure 0. 7. 15. A dialogue box then appears on the screen (as

shown in Figure 0. 7. 16).

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Applying Technology

Linear regression with a spreadsheet and calculator. After deciding on a number of

options, we obtain the scatter diagram shown in Figure 0. 7. 17. Bt selecting Add

Trending from the Chart menu, we can display the least-squares line and its

equation, 0.0026 51.399y x as shown in Figure 0. 7. 18.

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