Math Analysis Chapter 1 Notes: Functions and Graphsteachers.sduhsd.net/chayden/Chapter 1 Packethayden.pdfMath Analysis Notes Mr. Hayden 1 Math Analysis Chapter 1 Notes: Functions and

Math Analysis Notes Mr. Hayden

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Math Analysis

Chapter 1 Notes: Functions and Graphs

Day 6: Section 1-1 Graphs Points and Ordered Pairs

The Rectangular Coordinate System (aka: The Cartesian coordinate system)

Practice: Label each on the coordinate system shown. 1) x-axis, 2) y-axis, 3) origin 4) Quad I, 5) Quad II,

6) Quad III, 7) Quad IV. Also plot the points: A(−2. 4), B(4, −2), C(−3. 0), and D(0, −3).

Graphing Equations by Plotting Points

A relationship between two quantities can be expressed as an equations in two variables, such as: 2 4y x .

A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the

following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y

in the equation, we obtain a true statement.

Practice: In 1-2, Determine if the given ordered pairs are solutions to the equation 2 4y x .

1. (10, 96) 2. (0, 2)

Graphing an Equation Using the Point-Plotting Method.

1. Select values for x (for this section you will use the integers from −3 to 3)

2. Substitute each x-value into the equation and solve for y.

3. Create order pairs by grouping the x-value with it’s y-value.

4. Plot the order pairs


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Practice: 1. Graph 2 4y x using the Point-Plotting Method.

x 2 4y x Ordered Pair (x, y)

−3

−2

−1

0

1

2

3

Intercepts

x-intercept of a graph is the x-coordinate when the y-coordinate equals zero. We also can describe the

x-intercept graphically at the point where the graph intersects the x-axis.

y-intercept of a graph is the y-coordinate when the x-coordinate equals zero. We also can describe the

y-intercept graphically at the point where the graph intersect the y-axis.

Practice: 1-5, Identify the x- and y-intercepts of the given graphs or equation.

1. x-int: y-int: 2. x-int: y-int:

Plot all order pairs on a coordinate plane

and connect the points with a smooth curve if

and only if you are given an equation to

graph (which you are).

Do your homework on graph paper. If you

need graph paper you can get some by

printing off the net at

http://incompetech.com/graphpaper/plain/

The graph of 2 4y x

2 3 6x y 2 4y x


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3. x-int: y-int: 4. x-int: y-int:

5. 216y x

Day 7: Section 1-2 Basics of Functions and Their Graphs and Section 1-3 More on Functions and

There Graphs

Relation

Practice: Find the domain and the range of the relation: 5,12.8 , 10,16.2 , 15,18.9 , 20,20.7 25,21.8

Domain:

Range:

A relation of order pairs is a function if all of the domain (x) values are different. A function can have

two different domain values with the same range value.

Practice: In 1-2 Determine whether each relation is a function:

1. 1,2 , 3,4 , 5,6 , 5,8 2. 1,2 , 3,4 , 6,5 , 8,5

2 2 6 6 14 0x y x y 3 3 2y x x

Definition of a Relation

A relation is any set of ordered pairs. The set of all x-components of the ordered pairs is called the domain of the relation

and the set of all y-components is called the range of the relation.

Definition of a Function

A function is a correspondence from a first set, called domain, to a second set, called the range, such that each element in the

domain corresponds to exactly one element in the range.


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Determining Whether an Equation Represents a Function.

o To determine whether an equation defines y as a function of x:

1. Solve the equation in terms of y.

If only one value of y can be obtained for a given x, the equation is a function.

Example: x2 + y = 4

−x2 −x

2

y = 4 – x2

If two or more values of y can be obtained for a given x, the equation is not a function.

Example: x2 + y

2 = 5

−x2 −x

2

y2 = 5– x

2

2 25y x

25y x

Practice: In 1-2, determine whether each equation defines y as a function of x.

1. 4y x 2. 4x y

Function Notation

If an equation in x and y gives only one value of y for each value of x, then the variable y is a function of the

variable x. We use function notation by replacing y with f x . We think of a functions domain (x-

components) as the set of the function’s input values and the range (y-components) as the set of the function’s

output values. The special notation f x , read “f of x” represents the value of the function at the number

x.

x f(x) 2 3f x x

Evaluating a Function

To evaluate a function substitute the input value in for x and evaluate the expression.

Practice: If 2 2 7( )f x x x , evaluate each of the following:

(a) 5( )f (b) 4( )f x (c) ( )f x

From this last equation we can see that for each value of x, there

is one and only one value of y. For example, if x = 3, then

y = 4 – 32 = −5. This equation defines y as a function of x.

The ± in the last equation shows that for certain values of x,

there are two values of y. For example x = 2, then

25 (2) 5 4 1 1y . For this

reason, the equation does not define y as a function of x.

Input Output We read this equation as “f of x equal 2x +3”


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Graphs of Functions

Practice: Graph the functions 2( )f x x and 2 3( )g x x in the same rectangular coordinate system. Select

integers for x, starting with −2 and ending with 2. How is the graph of g related to the graph of f?

The Vertical Line Test

Practice: In 1-4, Use the vertical line test to identify graphs in which y is a function of x.

1. 2.

If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

If any vertical line intersects a graph in none or only at one point, , the graph does define y as a function of x.


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3. 4.

Section 1-3 More on Functions and Their Graphs

The Difference Quotient

To evaluate the difference quotient:

1. Find the value of ( )f x h by using substitution, replacing x with (x + h) and simplify the expression.

2. Use the expression found in step one and subtract ( )f x from it.

3. Use the expression found in step two and divide by h. This new expression is the called the difference

quotient.

Example: If 23 2 4( )f x x x , find and simplify each expression.

(a) ( )f x h (b) 0( ) ( )

,f x h f x

hh

(a) We find ( )f x h by replacing x with x + h

each time that x appears in the equation.

f(x) = 3x2 − 2x + 4

f(x + h) = 3(x + h)2 − 2(x + h) + 4

= 3(x2 + 2xh + h

2) – 2x – 2h + 4

= 3x2 + 6xh +3 h

2 – 2x – 2h + 4

The expression: ( ) ( )f x h f x

h

for h ≠ 0 is called the difference quotient.

Replace x with x + h

(b) Using our result from part (a), we obtain the

following:

2 2 23 6 3 2 2 4 3 2 4f x h f x x xh h x h x x

h h

( ) ( ) ( )

=2 2 23 6 3 2 2 4 3 2 4x xh h x h x x

h

= 2 2 23 3 2 2 4 4 6 3 2x x x x xh h h

h

( ) ( ) ( )

=26 3 2xh h h

h

=6 3 2h x h

h

( ) = 6x + 3h – 2

This is f(x+h) form part (a)

This is

f(x) from

the given

equation.


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Practice: In 1-2, Find and simplify the difference quotient: 0( ) ( )

,f x h f x

hh

for the given function.

1. 1

2( )f x

x 2. 22 3 1( )f x x x

Piecewise Functions

To evaluate a Piecewise Function

Only substitute the x-value into the expression where the inequality of x is true. When you evaluate this expression the

result is the value of the function with that given x.

Practice: In 1-2, Evaluate each piecewise function at the given values of the independent variable.

1. 3 5 if 0

4 7 if 0( )

x xf x

x x

(a) 2( )f (b) 0( )f (c) 3( )f

A function that is defined by two (or more) equations over a specified domain is called a piecewise function.

2.

2 9 if 3

3

6 if 3

( )

xx

h x x

x

(a) 5( )h (b) 0( )h (c) 3( )h


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Practice: In 1-2, State the intervals on which the given function is increasing, decreasing, or constant.

1. 2.

Increasing: Increasing:

Decreasing: Decreasing:

Constant: Constant:

Increasing, Decreasing, and Constant Functions.

1. A function is increasing on an open interval, I, if for any x1 and x2 in the interval where x1 < x2, then f(x1) < f(x2).

2. A function is decreasing on an open interval, I, if for any x1 and x2 in the interval where x1 < x2, then f(x1) > f(x2).

3. A function is constant on an open interval, I, if for any x1 and x2 in the interval where x1 < x2, then f(x1) = f(x2).

Increasing Decreasing Constant

x

y y

x x

y


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Relative Maxima and Relative Minima

Practice: Use the graph of f to determine each of the following. Where applicable, use interval notation.

1. the domain of f

2. the range of f

3. the x-intercept(s)

4. the y-intercept(s)

5. interval(s) on which f is increasing

6. interval(s) on which f is decreasing

7. intervals(s) on which f is constant

8. the relative minimum of f

9. the relative maximum of f

10. f(−3)

11. the value(s) of x when f(x) = −2

Definitions of Relative Maximum and Relative Minimum

1. A function value f(a) is a relative maximum of f if

there exists an open interval above a such that

( ) ( )f a f x for all x in the open interval. “Relative

Max is the point on top of a hill”

2. A function value f(b) is a relative minimum of f if

there exists an open interval about b such that

( ) ( )f b f x for all x in the open interval.

“Relative Min is the point at the bottom of a valley”

x

y

Relative Maximum

Relative Minimum


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Even and Odd Functions and Symmetry

Practice: In 1-3, Determine whether each of the following functions is even, odd, or neither.

1. 2 6( )f x x 2. 37( )f x x x 3. 5 1( )f x x

The function f is an even function if:

( ) ( )f x f x for all x in the domain of f.

The right side of the equation of an even function does not change if x is replaced with −x.

The function f is an odd function if:

( ) ( )f x f x for all x in the domain of f.

The right side of the equation of an odd function changes its sign if x is replaced with −x.

If you replace x with −x and you get something else that happens then we say the function is neither odd or even.

Even Functions and y-Axis Symmetry

The graph of an even function in which f(−x) = f(x) is

symmetric with respect to the y-axis.

f(x)=x2 −4

Odd Functions and Origin Symmetry

The graph of an odd function in which f(−x) = −f(x) is

symmetric with respect to the origin.

f(x)=x3


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Day 8: Section 1-4 Linear Functions and Slope; Section 1-5 More on Slope The Slope of a Line

Practice: In 1-2, Find the slope of the line passing through each pair of points:

1. (−3, 4) and (−4, −2) 2. (4, −2) and (−1, 5)

Possibilities for a Line’s Slope

Positive Slope Negative Slope Zero Slope Undefined Slope

Line Rises from left to right

Line falls from left to right

Line is horizontal

Line is vertical

Equations of Lines

1. Point-slope form: y – y1 = m(x – x1) where m = slope, (x1, y1) is a point on the line

2. Slope-intercept form: y = mx + b where m = slope, b = y-intercept

3. Horizontal line: y = b where b = y-intercept

4. Vertical line: x = a where a = x-intercept

5. General form: Ax + By + C = 0 where A, B and C are integers and A must be positive.

Practice: In 1-6, use the given conditions to write an equation for each line in (a) point-slope form, (b) slope-intercept form and (c)

General form.

1. Slope = 8, passing through (4, −1) 2. Slope = 2

3, passing through

3 7

4 8,

Definition of Slope

The slope of the line through the distinct

points 1 1,x y and 2 2

,x y is:

2 1

2 1

Change in y Rise

Change in x Run

y ym

x x

Where x2 – x1 ≠ 0.

Rise:

y2 − y1

Run:

x2 – x1

m > 0

m < 0

m = 0

m is

undefined


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3. Passing through (−3, −2) and (3, 6) 4. Passing through 1 2 15 7

and 2 3 4 6

, ,

5. x-intercept = 3 and y-intercept = 1 6. x-intercept = 2

3 and y-intercept =

5

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Parallel and Perpendicular Lines

Practice: Write an equation of the line passing through (−2, 5) and parallel to the line whose equation is y = 3x + 1. Express the

equation in point-slope from and slope-intercept form.

Practice:

Slope and Parallel Lines

1. If two nonvertical lines are parallel, then they have the same slope.

2. If two distinct nonvertical lines have the same slope, then they are parallel.

3. Two distinct vertical lines, both with undefined slopes, are parallel

If two lines are parallel

they have equal slopes:

m1 = m2

Slope and Perpendicular Lines

1. If two nonvertical lines are perpendicular, then the product of their slopes is −1.

2. If the product of the slopes of two lines is −1, then the lines are perpendicular.

3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.

If two lines are

perpendicular they have

slopes that are negative

reciprocals of each other:

1

2

1m

m

1. Find the slope of any line that is perpendicular to

the line whose equation is x + 3y – 12 = 0.

2. Write the equation of the line passing through

(−2, −6) and perpendicular to the line whose equation

is x + 3y – 12 = 0. Express the equation in general

form.


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Day 9: Section 1-6 Transformations of functions

Algebra’s Common Graphs

Constant Function

Domain: ,

Range: the single number “c”

Constant on ,

Even function (symmetric to y-

axis)

f(x) = c

Identity Function

Domain: ,

Range: ,

Increasing on ,

Odd function (symmetric to

Origin)

Absolute Value Function

Domain: ,

Range: 0,

Decreasing on 0, and

Increasing on 0,


axis)

f(x) = x

f(x) = x

Standard Quadratic Function

Domain: ,

Range: 0,

Decreasing on 0, and

Increasing on 0,


axis)

Standard Square Root Function

Domain: 0,

Range: 0,

Increasing on 0,

Neither even nor odd

f(x) = x2

f(x) = x

Standard Cubic Function

Domain: ,

Range: ,

Increasing on ,


Origin)

f(x) = x3


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Summary of Transformations (In each case, c represents a positive real number.)

To Graph Draw the Graph of f and: Changes in the Equation of y=f(x)

Vertical Shifts

y = f(x) + c

y = f(x) – c

Raise the graph of f by c units

Lower the graph of f by c units

c is added to f(x)

c is subtracted from f(x)

Horizontal Shifts

y = f(x + c)

y = f(x – c)

Shift the graph of f to the left c units

Shift the graph of f to the right c units

x is replaced with x + c

x is replaced with x − c

Reflection about the x-axis

y = −f(x)

Reflect the graph of f about the x-axis f(x) is multiplied by −1

Reflection about the y-axis

y = f(−x)

Reflect the graph of f about the y-axis x is replaced with −x

Vertical Stretching or Shrinking

y = cf(x), c > 1

y = cf(x), 0 < c < 1

Multiply each y-coordinate of y = f(x) by

c, vertically stretching the graph of f.

Multiply each y-coordinate of y = f(x) by

c, vertically shrinking the graph of f.

f(x) is multiplied by c, c > 1

f(x) is multiplied by c, 0 < c < 1

Horizontal Stretching or Shrinking

y = f(cx), c > 1

y = f(cx), 0 < c < 1

Divide each x-coordinate of y = f(x) by c,

horizontally shrinking the graph of f.

Divide each x-coordinate of y = f(x) by c,

horizontally stretching the graph of f.

x is replaced with cx, c > 1

x is replaced with cx, 0 < c < 1

Standard Cube Root Function

Domain: ,

Range: ,

Increasing on ,


Origin


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Practice: Graph the standard function f(x) and then graph the given function. Describe the transformations need to change the

common function f(x) to get g(x).

1. g(x) = −(x + 3)2 + 4 2. g(x) = 4x

Day 10: Section 1-7 Combinations of Functions and Composite Functions; Section 1-8 Inverse

Functions

Finding the Domain of a Function

Practice: In 1-3, Use interval notation to express the domain of each function:

1. 2 3 10( )f x x x 2. 2

5

7( )

xf x

x

3. 9 27( )h x x

Description of transformations

Description of transformations

The numbers excluded from a functions domain are real numbers that cause division by zero and real numbers that result in a

square root of a negative number.


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Practice: Let 5( )f x x and 2 1( )g x x . Find each of the following functions and determine the domain:

1. (f + g)(x) 2. (f – g)(x) 3. (fg)(x) 4. f

xg

Composite Functions

Practice: Given f(x) = 5x + 6 and g(x) = 2x

2 – x – 1, find each of the following composite functions:

1. ( )( )f g x 2. ( )( )g f x 3. 3( )( )f g

The Algebra of Functions: Sum, Difference, Product and Quotient of Functions

Let f and g be two functions. The sum f + g, the difference f – g, the product fg, and the quotient f

gare functions whose

domains are the set of all real numbers in common to the domains of f and g defined as follows:

1. Sum: ( ) ( ) ( )f g x f x g x

2. Difference: ( )( ) ( ) ( )f g x f x g x

3. Product: ( ) ( ) ( )fg x f x g x

4. Quotient: ( )

( )

f f xx

g g x

, provided g(x) ≠ 0

The Composition of Functions

The composition of the function f with g is denoted by f g and is defined by the equation:

( )( ) ( )f g x f g x .

The domain of the composite function f g is the set of all x such that

1. x is the domain of g and

2. g(x) is in the domain of f.


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Inverse Functions

Finding the Inverse of a Function

The equation for the inverse of a function f can be found as follows:

1. Replace f(x) with y in the equation for f(x).

2. Interchange x and y.

3. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and

his procedure ends. If this equation does define y as a function of x, the function f has an inverse function.

4. If f has an inverse function, replace y in step 3 with 1( )f x . We can verify our result by showing that 1f f x x

and 1f f x x .

Practice: Find the inverse of each function.

1. 2 7( )f x x 2. 34 1( )f x x 3. 3

1( )f xx

Definition of the Inverse of a Function

Let f and g be two functions such that

f g x x for every x in the domain of g

and

g f x x for every x in the domain of f.

The function g is the inverse of the function f and is denoted by 1f (read “f inverse”). Thus, 1f f x x and

1f f x x . The domain of f is equal to the range of 1f , and vice versa.

4. Verify that the

inverse function

found in problem 1

is correct.


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The Horizontal Line Test and One-to-One Functions

Practice: In 1-4 Which of the following graph represent functions that have inverse functions?

1. 2. 3. 4.

Graph of f and f

−1

To graph an inverse function given the graph of ordered pairs of the function

1. Find an ordered pair on the function.

2. To graph the inverse just take the x-coordinate of f(x) is the y-coordinate of f−1

(x) and the y-coordinate of f(x) is the x-

coordinate of f−1

(x).

3. Continue finding order pairs on f(x) and interchange the x- and y-coordinates to plot points on f−1

(x).

4. Connect points with a smooth curve.

Practice: Use the graph of f to draw the graph of its inverse function.

The Horizontal Line Test For Inverse Functins

A function f has an inverse that is a function f−1

, if there is no horizontal line that intersects the graph of the function f at more

than one point. If the function passes the Horizontal Line Test the function is said to be one-to-one.

1.

2.

3.

4.


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Day 11: Section 1-9 Distance and Midpoint Formulas; Circles; Section 1-10 Modeling with Functions

The Distance Formula

Practice: Find the distance between the two points given.

1. (−4, 9) and (1, −3) 2. 2 3 6 and 3 5 6, ,

The Midpoint Formula

Practice: Find the midpoint of the line segment with endpoints at: 7 7 5 11

and 5 15 2 2

, ,

.

Circles

Our goal is to translate a circle’s geometric definition into an equation.

The distance , d, between the points 1 1,x y and 2 2

,x y in a rectangular coordinate system is:

2 2

2 1 2 1d x x y y

To compute the distance between two points, find the square of the difference between the x-coordinates plus the square of

the difference between the y-coordinates. The principal square root of this sum is the distance.

Consider a line segment whose endpoints are 1 1,x y and 2 2

,x y . The coordinates of the segment’s midpoint are:

1 2 1 2

2 2,

x x y y

To find the midpoint, take the average of the two x-coordinates and the average of the two y-coordinates.

Geometric Definition of a Circle A circle is the set of all points in a plane that are equidistant from a fixed point, called the center. The fixed distance from

the circle’s center to any point on the circle is called the radius.

The Standard Form of the Equation of a Circle

The standard form of the equation of a circle with center (h, k) and radius r is:

2 2 2x h y k r


20

Practice: Write the standard form of the equation of the circle with the given information.

1. Center (−3, 1), r = 6 2. Endpoints of it’s diameter: (2, 3) and (−2, −1)

Converting the General Form of a Circle’s Equation to Standard Form and Graphing the Circle

To Convert General Form to Standard Form of a Circle

1. Group “like components” together and move constant term to other side.

2. Complete the Square for both x and y components. Remember to add the perfect square to both sides of the equal sign.

3. Factor each group on the left side of the equal sign to a square of a binomial. Your equation of a circle should now be in

standard form

Practice:

In 1-2, Write in standard form and graph:

1. x2 + y

2 + 6x + 2y + 6 = 0 2. x

2 + y

2 + 3x – 2y – 1 = 0

The General Form of the Equation of a Circle

2 2 0x y Dx Ey F

where D, E, and F are real numbers.


21

Modeling with Functions; Word Problems

Practice:

1. A car rental agency charges $200 per week plus $0.15 per mile to rent a car.

(a) Express the weekly cost to rent the car, f, as a function of the number of miles driven during the week, x.

(b) How many miles did you drive during the week if the weekly cost to ret the car was $320?

2. The bus fare in a city is $1.25. People who use the bus have the option of purchasing a monthly coupon book for $21.00. With

the coupon book, the fare is reduced to $0.50.

(a) Express the total monthly cost to use the bus without a coupon book, f, as a function of the number of times in a month

the bus is used, x.

(b) Express the total monthly cost to use the bus with a coupon book, g, as a function of the number of times in a month

the bus is used, x.

(c) Determine the number of times in a moth the bus must be used so that the total monthly cost without the coupon book

is the same as the total monthly cost with the coupon book. What will the monthly cost for each option?

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