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1.2 Functions and Their Properties PreCalculus

Unit 2 - 1

1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function given an set of numbers, an equation, or a graph 3. Describe the type of discontinuity in a graph as removable or non-removable 4. For a given function, describe the intervals of increasing and decreasing 5. Label a graph as bounded above, bounded below, bounded, or unbounded. 6. Find all relative extrema on a graph 7. Understand the difference between absolute and relative extrema 8. Describe the symmetry of a graph as odd or even This section is full of vocabulary (obvious from the list above?). We will be investigating functions, and you will need to answer questions to determine how each of these properties are applied to various functions. Function Definition and Notation In the last chapter, we used the phrase “y is a function of x”. But what is a function? In Algebra 1, we defined a function as a rule that assigned one and only one (a unique) output for every input. We called the input the domain and the output the range. Usually, the set of possible x-values is the domain, and the resulting set of possible y-values is the range. Definition: Function A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. To determine whether or not a graph is a function, you can use the vertical line test. If any vertical line intersects a graph more than once, then that graph is NOT a function. Example 1: A relation is ANY set of ordered pairs. State the domain and range of each relation, then tell whether or not the relation is a function. a) ( ) ( ) ( ){ }3,0 , 4, 2 , 2, 6− − b) ( ) ( ) ( )( ){ }4, 2 , 4,2 , 9, 3 9, 3− − − − c) d)

x

y

x

y


Unit 2 - 2

For many functions, the domain is all real numbers, or . We typically start with this, and then see if there are any values of x that cannot be used. The domain of a function can be restricted for 3 reasons that you need to be aware of in this course:

1. NO negatives … no negative numbers inside a square root

2. **NO zeros

… no zeros in the denominator of a fraction

3. log NO negatives ... and NO zeros . … no negatives and no zeros inside a logarithm

Example 2: Without using a graphing calculator, what is the domain of each of the following functions?

a) y = log x b) ( )2 9

3xh xx−

=+

c) 2

212

xyx x

+=

− −

Continuity In non – technical terms, a function is continuous if you can draw the function “without ever lifting your pencil”. Example 3: Tell whether the following graphs are continuous or discontinuous. If discontinuous, label each type of discontinuity as removable or non-removable.

x

y

x

y

x

y

x

y


Unit 2 - 3

Increasing/Decreasing Functions While there are technical definitions for increasing and decreasing, just remember to read the graph LEFT to RIGHT. Example 4: Using the graph below, what intervals is the function increasing? Decreasing? Constant? Boundedness You need to understand the difference between the following terms: BOUNDED BELOW: BOUNDED ABOVE: BOUNDED: Local and Absolute Extrema Extrema is the plural form of one extreme value. Extrema is one word that includes maximums and minimums. Local (or Relative) Extrema are y-values bigger (or smaller) than ____________________________________________. Absolute Extrema are y-values bigger (or smaller) than _________________________________________. Example 5: Suppose the following function is defined on the interval [a, b]. Label f (a) through f (e) as relative or absolute extrema.

x

y

a b c d e


Unit 2 - 4

Symmetry The next topic we concern ourselves with when dealing with functions is the idea of symmetry. Symmetry on a graph means the functions “look the same” on one side as it does on another. We are most concerned with the types of symmetry that can be explored numerically and algebraically in terms of ODD and EVEN functions. 3 Types of Symmetry Symmetry with respect to the y-axis: EVEN FUNCTIONS Graphically Numerically Algebraically Symmetry with respect to the origin: ODD FUNCTIONS Graphically Numerically Algebraically Symmetry with respect to the x-axis: NOT A FUNCTION Graphically Numerically Algebraically


Unit 2 - 5

1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 9. Prove a function is odd, even, or neither 10. Given a rational equation, find vertical asymptotes and holes (if they exist) 11. Given a rational equation, find the end behavior model 12. Given a rational equation, describe the end behavior using end behavior and limit notation. Odd vs Even Functions While there are many functions out there that are neither even nor odd, your concern with odd and even functions is twofold … #1: Identify graphs of functions that are Odd or Even #2: PROVE a function is Odd or Even While the first item above can be done graphically, numerically, or algebraically, the second is done ONLY algebraically. Example 6: Prove each function is odd, even, or neither. a) ( ) 25 5f x x= + b) ( ) 2 8 12g x x x= + + Vertical Asymptotes

Example 7: Graph the function ( ) 13

f xx

=+

. What happens at x = –3? … Why?

Example 8: Graph the function ( )2 9

3xg xx−

=+

. What happens at x = –3? … Why?

Vertical asymptotes occur in rational functions when __________________________________________________. If you plug a point into your function and get _________, you should look for a removable discontinuity … a.k.a. “a hole”.


Unit 2 - 6

End Behavior As x → ±∞, we say the end behavior of the function is a description of what value f (x) approaches. For now, we will focus on finding end behavior using a graphical approach. Example 9: Describe the end behavior of each graph shown below. We can write the end behavior in a more compact form using limit notation. There is a specific (a.k.a. correct) way of writing limits that you should learn. Example 10: Rewrite the given end behavior for f (x) using limit notation. Sketch a possible graph for f (x). a) As , ( )x f x→ ∞ → −∞ b) As , ( )x f x→ ∞ → ∞

As , ( )x f x→ −∞ → ∞ As , ( ) 0x f x→ −∞ → IF f (x) → “a number” as x → ±∞ … meaning the end behavior of the function is a number, we say the function has a horizontal asymptote (at that number) … more on this in section 2.7.

1.3 Twelve Basic Functions PreCalculus

Unit 2 - 7

1.3 TWELVE BASIC FUNCTIONS Lesson Targets for 1.3 1. Graph and Identify all 12 parent functions 2. Graph a piecewise function Parent Function #1: Linear Function (book refers to this as the identity function): Equation: ____________

Graph this function (label 5 points)

Domain: Range:

Symmetry:

Boundedness: Asymptotes: Discontinuities: Increasing/Decreasing: Extrema: End Behavior:

Parent Function #2: Quadratic Function (book refers to this as the squaring function): Equation: _____________


Domain: Range:

Symmetry:


x

y

x

y


Unit 2 - 8

Parent Function #3: Cubic Function (the book refers to this as the cubing function) : Equation: _____________


Domain: Range:

Symmetry:


Parent Function #4: Inverse Linear Function (book refers to this as the Reciprocal Function): Equation: _____________

Graph this function (label 2 points, a H.A., and a V.A.)

Domain: Range:

Symmetry: Boundedness: Asymptotes: Discontinuities: Increasing/Decreasing: Extrema: End Behavior:

x

y

x

y


Unit 2 - 9

Parent Function #5: Square Root Function: Equation: _____________


Domain: Range:

Symmetry: Boundedness: Asymptotes: Discontinuities: Increasing/Decreasing: Extrema: End Behavior:

Parent Function #6: Exponential Function: (the book uses only base e) We will use the equation xf x b ,

where b > 1 represents ____________, and 0 < b < 1 represents _______________.

Graph this function (label 2 points and a H.A.)

Domain: Range:

Symmetry:


x

y

x

y


Unit 2 - 10

Parent Function #7: Logarithm Function: (the book only uses a natural logarithm) We would like you to use logbf x x

Graph this function (label 2 points and a V.A.)

Domain: Range:

Symmetry:


Parent Function #8: Absolute Value Function: Equation: _____________


Domain: Range:

Symmetry:


x

y

x

y


Unit 2 - 11

Parent Function #9: Greatest Integer Function: Equation: ____________

Graph this function (label at least 6 points)

Domain: Range:

Symmetry:


Parent Function #10: Logistic Function: Equation: _____________

Graph this function (label 1 point and 2 H.A.)

Domain: Range:

Symmetry:


x

y

x

y


Unit 2 - 12

We will spend a large amount of time with the next two during second semester. For now, the parent function is enough. Parent Function #11: Sine Function: Equation: y = sin(x)

Graph this function

Domain: Range:

Symmetry:


Parent Function #12: Cosine Function: Equation: y = cos(x)

Graph this function

Domain: Range:

Symmetry:


x

y

x

y

1.5 Graphical Transformations PreCalculus

Unit 2 - 13

1.5 GRAPHICAL TRANSFORMATIONS Learning Targets for 1.5 1. Be able to graph transformations of parent functions 2. Be able to adjust the function when there is a horizontal stretch/shrink AND a left/right movement. You MUST be able to graph the parent functions from 1.3 in order to successfully transform them. A transformation is a stretch/shrink, a reflection, or simple movement of a parent function horizontally or vertically. The general rule … “Inside … x’s … opposite” and “Outside … y’s … same” Example 1: Suppose you are given the function f (x). If a, b, c, and d are real numbers, our transformed function is

a f b x c d The box below summarizes our transformation rules:

Inside Outside

/

f x c

f x c

f x d

f x d

/

f bx

1bf x

f x

a f x

1a f x

f x

: Make sure you _______________________ before you ___________________________.

1.5 Graphical Transformations PreCalculus

Unit 2 - 14

Example 2: Perform the following transformations on the graph of f (x) below: a) 3 2f x

b) 2xf

c) 2 3f x Example 3: Graph the following functions by transforming their parent functions: **MOST OFTEN MISSED** a) 3 6f x x b) 3log 5g x x

x

y

x

y

x

y

Unit 2–15

4.4 and 4.5 Graphing Trigonometric Functions Pre-Calculus Learning Targets

1. Apply translations to the three parent trigonometric functions. 2. Given the attributes of a trigonometric function write the equation of a trig. function. 3. Given a graph, identify the attributes of the function and write its equation.

Before we begin, let’s review the Parent Sine and Cosine curves we learned about in both Algebra 2 and in our Parent Functions Lesson 1.3 in this unit. WARM UP: Label the scale for graphs below to identify 9 key points…what do you notice? Sine Function: ( ) sinf x x= Cosine Function: ( ) cosf x x= Here is some important terminology we use with trigonometric graphs.

• Periodic Functions: Any function that has a positive number p such that ( ) ( )f x p f x+ = for every value of x. In other words…any function whose graph repeats itself at equal intervals.

• Period: The smallest value of p. In other words…the length of x interval needed to complete one cycle of the graph.

• Amplitude: ½ (maximum – minimum) In other words…the height of the curve from the midline or x-axis.

What about Tangent? Tangent Function: ( ) tanf x x=

• Period:

• Amplitude:

• Vertical Asymptotes:

Unit 2–16

To review our translations…A number added inside effects the horizontal direction, but it’s backwards, and a number added outside effects the vertical direction. When we apply these transformations to the trig functions, we get…

[ ]sin ( )y a b x c d= − + or [ ]cos ( )y a b x c d= − +

• Amplitude:

• Period:

• Phase Shift:

• Vertical Shift:

Example 1: List the amplitude, period, phase shift and vertical shift. Then graph at least one period of the function. Be sure to label the scale of each axis.

a) 4sin 2y x= − + b) cos 12

y xπ

= + +

c) sin 33

y xπ

= +

d) tan 54x

y = −

For this last example, be careful…the phase shift is NOT 2π

!

e) 3sin 2 12

y xπ

= + +

Unit 2–17

Example 2: Write two equations of the cosine function whose amplitude is 2, period is 6π , phase shift is 23π

and vertical shift is –5. Example 3: Write the equation for the given graph using…

a) the Sine Function b) the Cosine Function.

4

2.3 Polynomial Functions of Higher Degree with Modeling PreCalculus

Unit 2 - 18

2.3 POLYNOMIAL FUNCTIONS OF HIGHER DEGREE WITH MODELING Learning Targets: 1. Be able to describe the end behavior of any polynomial using limit notation. 2. Understand how the multiplicity of a zero changes how the graph behaves when it meets the x-axis. 3. Use end behavior and multiplicity of zeros to sketch a polynomial by hand. Our focus today is on polynomials of degree 3 (cubic), degree 4 (quartic), or higher. End behavior describes what happens to the y – values of the function as x approaches positive or negative infinity as we learned in lesson 1.2. If you can remember the graphs of y = x, y = –x, y = x2, and y = –x2, then you can remember the end behavior of ALL polynomial functions using the “leading coefficient” and highest power. Example 1: Describe the end behavior of the polynomial function using lim ( )

xf x

→−∞ and lim ( )

xf x

→∞. Confirm graphically.

a) ( ) 63 5 3f x x x= − + b) ( ) 3 27 8 9g x x x x= − + − + When the end behavior of a function goes to positive or negative infinity, we will write lim ( )

xf x

→∞= ∞ or – ∞, but since

infinity is not a real number, in Calculus we say the limit does not exist. From Unit 1:

• The zeros of a function are the x values that make f(x) = 0. These x values are called the x-intercepts when they are real numbers.

• We can use factoring to find these zeros algebraically.

• The multiplicity of each zero is the number of times the factor occurs in the factored form of the polynomial.

Even Power Odd Power

Positive Coefficient

Negative Coefficient

lim ( )x

f x→−∞

= lim ( )x

f x→∞

= lim ( )x

f x→−∞

= lim ( )x

f x→∞

=

lim ( )x

f x→−∞

= lim ( )x

f x→∞

= lim ( )x

f x→−∞

= lim ( )x

f x→∞

=

2.3 Polynomial Functions of Higher Degree with Modeling PreCalculus

Unit 2 - 19

Example 2: List each zero and state its multiplicity. Then, graph each function on your calculator. What do you notice? a) ( ) ( )23f x x x= − b) ( ) 2 3( 1)( 2) ( 4)h x x x x= + − −

At a zero with EVEN multiplicity, the graph of the function will _________________________________. At a zero with ODD multiplicity, the graph of the function will __________________________________. Putting it all together Example 3: State the degree and list the zeros of the polynomial function. State the multiplicity of each zero and whether the graph crosses the x-axis at the corresponding x-intercept. Using what you know about end behavior and the zeros of the polynomial function, sketch the function. a) ( ) 4 225f x x x= − Degree: Sketch: Zero(s) & Multiplicity:

End Behavior: b) ( ) ( )( )23 2 7h x x x= − − + Degree: Sketch: Zero(s) & Multiplicity:

End Behavior:

2.7 Graphs of Rational Functions PreCalculus

Unit 2 - 20

2.7 Graphs of Rational Functions – Day 1 Learning Targets: 1. Find the following attributes of rational functions:

i) End Behavior (including Horizontal Asymptotes or Slant Asymptotes) DAY 1 ii) Vertical Asymptotes (distinguish between holes and vertical asymptotes) DAY 1 iii) x-intercept(s) DAY 2

iv) y-intercept DAY 2 2. Graph a rational function by hand including all of the above attributes (if they exist). DAY 2

All of the functions we will deal with for the remainder of this chapter will be Rational Functions. Let’s begin with a definition and a review of end behavior…

Rational Function: ( )

( )

f xy

g x=

Example 1: Find the end behavior of the function using limit notation.

NOTE: For ALL rational functions, IF the end behavior is a numeric value, that value will be the SAME as x → ±∞ . In addition, the end behavior for all rational functions can fit into three categories. Examples are provided below.

A) ( ) 2 5 4xf x

x x=

− + B) ( )

3

3

427

xg xx

=−

C) ( )2 4

2 5xh xx−

=+

Example 2: Find the end behavior for each rational function above. a) b) c) Example 3: Since end behavior only uses the 1st term of a polynomial and a rational function is simply the ratio of two polynomials, we can use the ratio of the end behaviors to find the end behavior of the rational function. How does the ratio of the end behaviors compare to the end behavior of the graphs?


Unit 2 - 21

Here’s why this works…

Step 1: Consider the parent function ( ) 1h xx

= . As x →∞ , where does h (x) approach?

Step 2: Let’s rewrite ( )3

3

427

xg xx

=−

…

We need to put all this information together with the what we already know about Domain, Vertical Asymptotes and Holes from previous chapters…yep, you were supposed to remember that!!

Example 4: Use the equation below to answer the following questions: 2

2

2 182 11 15

xh xx x

a) Find the end behavior. Include HA or slant asymptotes. b) Find any vertical asymptotes. Distinguish between holes and vertical asymptotes.

Example 5: Use the equation below to answer the following questions: 2 6 8

1x xg x

x

a) Find the end behavior. Include HA or slant asymptotes. b) Find any vertical asymptotes. Distinguish between holes and vertical asymptotes.


Unit 2 - 22

2.7 Graphs of Rational Functions – Day 2 Learning Targets: 1. Find the following attributes of rational functions: iii) x-intercept(s) DAY 2

iv) y-intercept DAY 2 2. Graph a rational function by hand including all of the above attributes (if they exist). DAY 2

An x-intercept is any x-value whose y-value = ___________. We will write our answer as _____________. A y-intercept is any y-value whose x-value = ___________. We will write our answer as _____________. Example 1: Find the x-intercept and the y-intercept of ( ) 4 7f x x= − . Let’s see how these definitions apply to Rational Functions…

• When does a fraction = 0?

• How do we find the y-value of a fraction when x=0?

Example 2: Find the x-intercept and the y-intercept of 2 7 10

( )4

x xf x

x

− +=

+

Now to put this information with what we learned about asymptotes in 2-7 day 1. We will revisit the same examples.

Example 3: Given 2

2

2 18 2( 3)( 3)(2 5)( 3)2 11 15

x x xh xx xx x

a) Find the end behavior. End Behavior: HA: y = 1

b) Find any vertical asymptotes. Hole at x = 3; VA: x = 5

2

c) Find the x-intercept(s). d) Find the y-intercept. e) Graph the function using the information above and any additional points as needed.

x

y

lim ( ) 1x

h x→±∞

=


Unit 2 - 23

For examples 4-5, for the given function find the following (if they exist). a. End Behavior including the equations of horizontal or slant asymptotes b. Vertical Asymptote(s). Distinguish between holes and vertical asymptotes. c. x- intercept(s) d. y-intercept and additional points in each region to determine the shape. e. Graph without a calculator.

Example 4: 2 6 8 ( 4)( 2)

1 1x x x xg x

x x

a) Find the end behavior. End Behavior: SA: 5y x= +

b) Find any vertical asymptotes. VA: x = 1

Example 5: 2

3 12 3

xk xx x

x

y

x

y

lim ( )

lim ( )x

x

g x

g x→∞

→−∞

= ∞

= −∞

1.4 Building Functions from Functions PreCalculus

Unit 2 - 24

1.4 BUILDING FUNCTIONS FROM FUNCTIONS Learning Targets 1. Be able to Add, Subtract, Multiply, and Divide two functions 2. Be able to Compose two functions 3. Find an Inverse Functions Graphically, Numerically, and Algebraically 4. Prove (or Verify) two functions are Inverses Creating functions from other functions can be done in a variety of ways. We are going to add, subtract, multiply, divide, and compose functions to create new functions. Example 1: Express in function notation what the following expressions mean:

a) ( )( )f g x+ b) ( )( )f g x− c) ( )( )fg x d) ( )f xg

Example 2: Consider the two functions ( ) 6f x x= − and ( ) 2g x x= + . What is the Domain of each? a) Find ( )( )g f x− . State the Domain of the new function.

b) Find ( )f xg

. State the Domain of the new function.

Composite Functions When the range of one function is used as the domain of a second function we call the entire function a composite function. We use the notation f g x f g x to describe composite functions. This is read as "f composed with g" or "f of g of x". The function f in the example above is an example of a composite function. The linear function “x + 2” is applied first, then the square root function. Example 3: Suppose 21f x x and g x x .

a) Find g f x . What is the domain of g f x ? b) Find f g x . What is the domain of f g x ?


Unit 2 - 25

Inverse Functions A function has an inverse function if and only if the original function passes the Horizontal Line Test. The Horizontal Line Test works just like the Vertical Line Test (it’s just horizontal ). All an inverse function does is switch the x and y or the domain and range. Example 4: The graphs of two functions are shown below. Do they have an inverse? Why or why not? Once we know whether a function has an inverse, our next task is to find an equation and/or a graph for the inverse. Finding the Inverse Graphically: Reflect the graph of the original function over the line y = x. Finding the Inverse Numerically: Plot the reverse of the coordinates. Finding the Inverse Algebraically (… THIS is our focus): Switch the x and y in the original equation, then solve the new equation for y in order to write y as a function of x. Your NEW y can be written as 1f x Example 5: Let 3 1f x x , find the inverse algebraically.

Verifying Inverses It is one thing to find the inverse function (either graphically or algebraically), but it is another to verify that two functions are actually inverses. Whenever you are verifying anything in mathematics, you must go back and use the definition. Definition: Inverse Function A function f x has an inverse 1f x if and only if 1 1f f x x f f x Example 6: According to this definition, how many composite functions must be used to verify inverses? Example 7: Verify that the function you found in example 5 is in fact the inverse function.

2 5k x x x 3 1h x x


Unit 2 - 26

Example 8: Find 1f x and verify if 32

xf xx

. … YEAH … you’ll probably need most of this space .

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