Lesson 5.1: Introduction to Functions

Learning Goals:

1) What is a function and how do you determine if a relation is a function?2) What is a one-to-one function and how do you determine if a function is

one-to-one?3) How do you evaluate functions?

What is a function?

A relation is a set of ordered pairs or collection of points (i.e.: coordinates, equations, graphs, circle diagrams)

Based on your observations of the above table, what is a function?

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A function is a relation in which each element of the domain corresponds to one and only one element in the range or a relation in which no two ordered pairs have the same first element. It can also be called a relation with no repeating x- values.

To determine if a relation is a function given its graph we use the vertical line test.

Be sure students understand the definition of onto. A function is onto if each element of B is mapped by at least one element of the domain A.

Examples:

1. Consider the correspondence between a set of people in a room and his/her height in centimeters.

Barbara→165

Keith→176

Robert→165

Maria→168

Notice that two different people may have the same height, but no person can have 2 different heights.

This is an example of a function.

2. Decide if the following relations are functions or not and explain why.

(a) { (1 ,2 ) , (3 ,−1 ) , (5 ,7 ) , (−2 ,1 ) } Yes, no repeating x-values.

(b) {(0 ,2 ) , (3,2 ) , (−5 ,1 ) , (6 ,2 ) } Yes, no repeating x-values.

(c) {(−1 ,1) ,(2 ,3) ,(2 ,−3) ,(4 ,7)} No, x−¿value of 2 repeats.

(d) { (6 ,1 ) , (−2 ,3 ) , (7 ,2 ) , (−2,1 ) } No, x−¿value of −2 repeats.

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3. Determine if each of the following graphs are functions.

yes no no yes

One-to-One Functions

What is the difference between the functions in column A and those in column B?

A One-to-One Function: is a function with no repeating x∨ y-values.

To see if a relation is a one-to-one function use both the vertical line test (x-values) and the horizontal line test ( y−¿values)

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4. Determine if each of the following relations are one-to-one and explain.

a) {(2 ,1 ) , (3 ,2 ) , (4 ,2 ) , (5 ,1 ) } Not 1-1, but is a function.

b) {(1 ,−1 ) , (2 ,−2 ) , (3 ,−3 ) , (4 ,−4 ) } 1-1

c) {(4 ,1 ) , (5 ,0 ) , (4 ,2 ) , (3 ,−1 ) } Not 1-1, and not a function.

Determine if the following relations are one-to-one:

No (fails HLT), No (fails VLT), Yes, Yes

Evaluating Functions

The f (x) notation can be thought of as another way of representing the y-value in a function, especially when graphing. The y-axis is even labeled as the f (x) axis, when graphing. Often times, mathematics has a variety of ways to say the same thing. Examine these notations that are all equivalent:

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5. Write each of the following in function notation:

(a) The population of a certain city is represented by the equation

y=500,000 (2)t. f (t )=500,000(2)t

(b) The amount of money a school fundraises is represented by the equation

y=500+2n. g (n )=500+2n∨g (n )=500+2n

(c) The speed of a baseball hit by a batter is represented by

y=9.8v2−16 v+32. h(v )=9.8v2−16v+32

Practice:

6) If f ( x )=2x2+6 x, find f (−2).f ( x )=2 (−2 )2+6 (−2 )

f ( x )=2 (4 )−12f ( x )=8−12f (−2 )=−4

7) If g ( x )=√2x+22, find g(−3).g ( x )=√2 (−3 )+22

g ( x )=√−6+22g ( x )=√16g (−3 )=4

8) If f ( x )=4 x−7, find x when f ( x )=13.

13=4 x−720=4 x5=x

9) If g ( x )=3x+52 and g ( x )=4, find x.

4=3 x+52

8=3 x+53=3x1=x

10) If f ( x )=k x2 and f (2 )=3, find k3=k (2 )2

3=4 k34=.75=k

11) Find k when f ( x )=3 x2+kx−1 and f (1 )=9.

9=3 (1 )2+k (1 )−19=3+k−19=2+k7=k

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Homework 5.1: Introduction to Functions

1. Explain whether the pairing is a function.

2. Which graph does not represent a function.

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3. Which graph represents a function?

4. Which relation is not a function?

1) (x−2)2+ y2=4 2) x2+4 x+ y=4

3) x+ y=4 4) xy=4

5. Which diagram is not a function?

6. Explain whether or not each of the following is a one-to-one function.7

7. If g ( x )=−3 x+2 and g ( x )=−1, find x.

8. If f ( x )=6x2−10 x+2, find f (2).

9. If g ( x )=x2+2 and g ( x )=11, find the positive value of x.

10. Find k if y=3 k2 x when f (6 )=2.

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11. Given this graph of the function f ( x ) :

Find:

(a) f (−4 )=¿

(b) f (2 )=¿

(c) What is x when f ( x )=4?

(d) What is x when f ( x )=−1?

Lesson 5.2: Domain and Range of a Function

Learning Goals:

1) What is domain and range of a function?2) How do we determine the domain and range of a function?

Do Now:

Write an inequality that represents the solution set given by the shaded region of the number line.

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x>−7∧x<3 x≤ 13∨x≥2

−7<x<3 Or=∪

And=∩

Review of Interval Notation:

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1. Write each of the following solutions in set builder notation and interval notation.

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Set Builder Notation (inequality) Interval Notation [ or )x≥2 [2 , ∞ )x<8 (−∞,8)

x≤−3∨x ≥−1 ¿∨¿−4≤ x ≤−2 [−4 ,−2]x<2∨x>3 (−∞,2 )∨(3 , ∞)

The domain of a function is the set consisting of all first elements of the ordered pairs. (Abscissas or x- values.) “input” of the relation

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The range of a function is the set consisting of all second elements of the ordered pairs. (Ordinates or y-values.) “output” of the relation

Finding the domain and range from the GRAPH

2. Find the domain and range of each of the following in interval notation.

a) D= [−1 ,6 ]∧R=[−2 ,5] b) D= [−6 ,5 ]∧R=[−8 ,8]

c) D= (−∞ , ∞)∧R=[−8 , ∞ ) d) D=¿∧R=(−∞,∞ )

e) D= (−∞ ,∞)∧R=[−4 ,∞ )

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Finding the domain and range from the EQUATION

Use the calculator to graph and sketch!

3. Find the domain and range of each of the following.

a) y=−(x+5)2 b) y=|x+3|

D=(−∞,∞) Absolute Value!

R=¿ Math→ NUM →1 :|¿|

D= (−∞ ,∞)∧R=¿

2nd→0 :catalog →|¿|

c) y=√x+1−2 d) y=√x2−2 x−3

D= [−1 , ∞ ) D= (−∞ ,−1 ]∧¿

R=¿ R=¿

f) If the domain of f ( x )=2x+3 is {−3<x ≤0}, which number is not in the range?

(a) −1 (b) 0 (c) 3 (d) 6

Homework 5.2: Domain and Range of a Function

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1. What is the domain of the function f ( x )= x−4√6−x?

(1) { x≠ 4 , x≠6 } (2) { x|x≤6 } (3) { x|x≥6 } (4) { x|x<6 }

2. Function m(x) is defined as m (x )= x+5x2+3 x−10 . State the domain of function m(x).

3. What is the domain of the function h ( x )=√ x2−4 x−5?

(1) { x|x≥1∨x ≤−5 } (2) { x|x≥5∨x ≤−1 }

(3) { x|−1≤ x≤5 } (4) { x|−5≤ x ≤1 }

4. Which relation is not a function?

(1) y=2x+4 (2) y=x2−4 x+3 (3) x=3 y−2 (4) x= y2+2 x−3

5. Which graph does not represent a function?

6. State the domain and range of each of the following in interval notation:

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7. If h ( x )=√2x+3, find h(−1).

8. If f ( x )=−3x+10 and f ( x )=13, find x.

9. If h ( x )=−kx+14 and h (2 )=18, find the value of k

Lesson 5.3: Evaluating Functions and Operations with Functions

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Learning Goals:

1) How do you evaluate functions?2) How do you perform operations with functions?

Function notation:

We have seen linear functions written in the form y=mx+b. By naming a function f , you can write it in function notation.

f ( x )=mx+b

The symbol f (x) is another name for y and is read as “the value of f at x” or simply as “f of x It does not mean f times x.

Other letters used to represent functions are g or h. f ( x )= y

Directions: Evaluate the following functions:

1. f ( x )=x−7, when x=3 2. g ( x )=x2, when x=−5

f (3 )=3−7 g (−5 )=(−5)2

f (3 )=−4 g(−5)=25

3. f ( x )=x2−x+3 ; f (−2) 4. If f ( x )=kx+5 and f (3 )=36,

f (−2 )=(−2)2−(−2)+3 find the value of k. x=3∧ y=36

f (−2 )=4+2+3 36=k (3 )+5

f (−2 )=9 31=3 k

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=k

5. f ( x )=2x2−3 x+4 ; f (−x ) 6. f ( x )=2x+5 ; f ( x−8 ) .

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f (−x )=2 (−x )2−3 (−x )+4 f ( x−8 )=2 ( x−8 )+5

f (−x )=2x2+3 x+4 f ( x−8 )=2 x−16+5

Don’t try to solve/factor! f ( x−8 )=2 x−11

7. f ( x )=2x2+4 ; f (a−2) 8. f ( x )=x2−2x+4 ; f (x+2)

f ( a−2 )=2 ( a−2 )2+4 f ( x+2 )= (x+2 )2−2 ( x+2 )+4

f ( a−2 )=2 ( a−2 ) (a−2 )+4 f ( x+2 )= (x+2 ) ( x+2 )−2 x−4+4

f ( a−2 )=2 ( a2−4a+4 )+4 f ( x+2 )=x2+4 x+4−2x

f ( a−2 )=2a2−8a+8+4 f ( x+2 )=x2+2 x+4

f ( a−2 )=2a2−8a+12

Definition of Operations of Functions:

Sum ( f +g ) (x )=f ( x )+g (x)Difference (order matters) ( f −g ) ( x )= f ( x )−g (x)Product ( f ∙ g ) ( x )=f (x) ∙ g(x )Quotient (order matters) ( f

g ) ( x )= f (x)g (x)

, g(x )≠0

For each new function, the domain consists of those values of x common to the domains of f and g. The domain of the quotient function is further restricted by excluding any values that make the denominator, g(x ), zero.

9. Given f ( x )=x2−4 and g ( x )=x+2, find each function.

(a) ( f +g ) (x ) (b) ( f −g ) ( x )

f ( x )+g(x ) f ( x )−g (x)

(x¿¿2−4)+(x+2)¿ ( x2−4 )−(x+2)

x2+ x−2 x2−4−x−2

x2−x−6

(c) ( f ∙ g)(x) (d) ( fg )(x )

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f (x) ∙ g (x) f (x )g (x)

( x2−4 ) ( x+2 ) x2−4x+2 DOPs

x3+2x2−4 x−8(x−2)(x+2)

x+2

x−2

10. Given f ( x )=2x2+5x−3 and g ( x )=x+3, find each function.

(a) ( f +g ) (x ) (b) ( f −g ) ( x )

(2 x2+5 x−3 )+(x+3) (2 x2+5 x−3 )−(x+3)

2 x2+6 x 2 x2+5 x−3−x−3

2 x2+4 x−6

(c) ( f ∙ g)(x) (d) ( fg )(x )

(2 x2+5 x−3 ) ( x+3 ) 2x2+5 x−3x+3

2 x3+6 x2+5 x2+15 x−3 x−9(2 x−1)(x+3)

x+3

2 x3+11 x2+12x−9 2 x−1

11. Given g (a )=−3a2−a and h (a )=−2a−4, find each of the following:

(a) ( g+h ) (−5 ) (b) ( gh ) (4 )

(−3a2−a )+ (−2a−4 ) −3a2−a−2a−4

−3a2−3a−4−3 (4 )2−4−2 (4 )−4

−3 (−5 )2−3 (−5 )−4−3 (16 )−4

−8−4

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−3 (25 )+15−4 −48−4−12

−75+15−4 −52−12

−64 133

Find the indicated value:

(a) ( f +g ) (1 )=¿ (b) ( f −g ) (1 )=¿

−2+2 −2−2

0 −4

(c) ( f ∙ g ) (2 )=¿ (d) ( f +g)(−3)

(0∗2 ) −2−2

0 −4

Homework 5.3: Evaluating Functions and Operations with Functions

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Evaluate each of the following functions:

1. f ( x )=2x−3 ; f (−5 ) 2. f ( x )=x2−4 ; f (x+1 )

3. Given f ( x )=x2−5 x+6 and g ( x )=x−3, find each function

(a) ( f +g ) (x ) (b) ( f −g ) ( x )

(c) ( f ∙ g)(x) (d) ( fg )(x )

7. Find the indicated value:

(a) ( f −g)(−2) (b) ( f ÷ g)(2)

Lesson 5.4: Composition of Functions

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Lesson Goals:

1) What is composition of functions?2) How do we perform the composition of functions?

Warm-Up: You are shopping for sneakers. Foot locker is having a sale – everything in the store is 20% off. You find a pair of sneakers that are originally priced for $90. When you get to the register, you find out that there is an additional 15% off.

a) Find the cost of the sneakers with 20% off.

90∗.20=18 off so 90−18=72∨90 ( .80 )=72

b) Using your answer to part (a) find the cost of the sneakers with the additional 15% off.

72+.15=10.8 off so72−10.8=$61.20

In this example we see that the output of the 20% off is used as the input to the 15% off function. This is known as the composition of functions and can be generalized for any functions f and g.

$90→20%→72→15%→ $ 61.20

There are 2 notations we use for composition of functions:

Order matters! Always work right to left!

( f ∘ g ) (x )=f (g ( x ))

( g∘ f ) (x )=g ( f ( x ))

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Example 1: Given f ( x )=x2−5 and g ( x )=2 x+3, find the values for each of the following:

a) f (g (1 ))g (1 )=2 (1 )+3¿2+3¿5Then f (5 )=52−5¿25−5¿20

b) (g∘ f )(3)f (3 )=32−5¿9−5¿4Then g (4 )=2(4)+3¿8+3¿11

c) (g∘g)(0)g (0 )=2 (0 )+3¿3g (3 )=2 (3 )+3¿6+3¿9

d) ¿f (−1 )=(−1 )2−5¿1−5¿−4f (−4 )=(−4 )2−5¿16−5¿11g (11)=2 (11 )+3¿22+3¿25

Example 2: The graphs below are the functions y=f ( x )∧ y=g( x). Evaluate each of the following questions based on these two graphs.

a) g (f (2 ) ) b) f ( g (−1 ) ) c) g ( g (1 ) )

f (2 )=3 g (−1 )=−5 g (1 )=3

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g (3 )=3 f (−5 )=−0.5 g (3 )=3

d) ( g∘ f ) (−2 ) e) ( f ∘ g ) (0 ) f) ( f ∘ f ) (0 )

f (−2 )=1 g (0 )=0 f (0 )=2

g (1 )=3 f (0 )=2 f (2 )=3

Example 3: Given the function f ( x )=3 x−2∧g ( x )=5x+4, determine formulas in simplest y=ax+b form for:

a) f ( g ( x ) ) b) ( g∘ f ) (x )

g ( x )=5 x+4 f ( x )=3 x−2

f (5 x+4 )=3 (5 x+4 )−2 g (3 x−2 )=5 (3 x−2 )+4

¿15 x+12−2 ¿15 x−10+4

¿15 x+10 ¿15 x−6

Example 4: Given the function f ( x )=x2∧g ( x )=x−5, determine formulas in simplest form for:

a) ( f ∘ g)(x) b) g( f ( x ))

g ( x )=x−5 f ( x )=x2

f ( x−5 )=( x−5 )2 g ( x )=x2−5

¿ ( x−5 ) ( x−5 )

¿ x2−10 x+25

PUSH YOURSELF!

Example 5: For each function h below, find two functions f and g such that h ( x )= f (g ( x )). There are many correct answers.

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a) h ( x )=(3x+7)2 b) h ( x )= 3√ x2−8

g ( x )=3 x+7 g ( x )=x2−8

f ( x )=x2 f ( x )= 3√x

c) h ( x )= 4(2 x−3)3 d) h ( x )=(x+1)2+2(x+1)

g ( x )=2 x−3 g ( x )=x+1

f ( x )= 4x3

f ( x )=x2+2 x

Homework 5.4: Composition of Functions

1. If f ( x )=x2+4 and g ( x )=√1−x, what is the value of f (g (−3 ))?

2. Using f ( x )=x2 and g ( x )=x+1, find (g∘ f )(−1).

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3. If f ( x )=3 x+1 and g ( x )=x2−1, find ( f ∘ g)(2).

4. If g ( x )=3 x−5 and h ( x )=2 x−4, find the formula for (g∘h)(x ).

5. If f ( x )=x2+5 and g ( x )=x+4, find the formula for f (g ( x )).

6. The graphs of y=h(x) and y=k (x) are shown below. Evaluate the following based on these two graphs.

a) h(k (−2 )) b) (k ∘h)(0) c) h(h (−2 )) d) (k ∘ k )(−2)

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7. Consider the functions f ( x )=2x+9 and g ( x )= x−92 . Calculate the following:

a) g (f (15 )) b) g (f (−3 ) )

c) g( f ( x )) d) What appears to always be true when you compose these two functions?

Lesson 5.5: Inverse Functions

Learning Goals:

1) What is the inverse of a function and how do we find the inverse of a function?

2) Graphically, what is the relationship between a function and its inverse?

Warm-Up: Consider the two linear functions given by the formulas f ( x )=3 x+72

and g ( x )=2x−73 .

(a) Calculate f ( g (−1 ) ) (b) Calculate f ( g (5 ))

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g (−1 )=2 (−1 )−73

=−2−73

=−93

=−3 g (5 )=2 (5 )−73

=10−73

=33=1

f (−3 )=3 (−3 )+72

=−9+72

=−22

=−1 f (1 )=3 (1 )+72

=3+72

=102

=5

(c) Without calculation, determine the value of f (g (π )).

π Because f ( x )∧g (x) are inverses!

Inverse Functions

Two functions are inverses because they literally “undo” one another. The general idea of inverses, f ( x ) and g(x ), is shown below in the mapping diagram.

In other words, two functions are inverses if their x and y-values are switched.

If a function y=f (x ) has an inverse that is also a function we represent it as y=f −1(x ).

The inverse does not mean reciprocal so f−1(x)≠ 1f (x)

How to Find the Inverse of a Function

The process of finding an inverse is simply swapping the x and y coordinates.

1. Find the inverse of h ( x )= {(1 ,2 ) , (−3 , 4 ) , (5 ,−6 ) }

h−1 ( x )= {(−6 ,5 ) , (2 ,1 ) ,(4 ,−3)}

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2. Graph and label f−1(x), the inverse of f (x) on the set of axes below.

What type of reflection are the graphs of f (x) and its inverse?

Reflection over the line y=x

Solving for an inverse relation algebraically is a three step process:

1) Set the function ¿ y2) Swap the x and y variables3) Solve for y

3. What is the inverse of the function 4. What is the inverse of the function

g ( x )=3 x+5? f ( x )=−12

x−2?

y=3 x+5 y=−12

x−2

x=3 y+5 x=−12

y−2

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x−5=3 y x+2=−12

y

y= x−53

=g−1(x) −2 x−4= y=f −1(x)

5. What is the inverse of the function m (x )= 5+ x6−2 x?

y= 5+x6−2x

x= 5+ y6−2 y

6 x−2 xy=5+ y

6 x−5=2xy+ y

6 x−5= y (2 x+1 )GCF

6 x−52 x+1

= y=m−1(x )

6. When f ( x )= x−72 , what is the value of ( f ∘ f −1)(3)?

y= x−72

x= y−72

2 x+7= y=f −1(x )

( f ∘ f −1 ) (3 )=¿

( f−1 ) (3 )=2 (3 )+7=6+7=13

( f ) (13 )=13−72

=62=3

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How to Determine if the Inverse of a Graph is a Function

To find the inverse of a function, simply switch the x and ycoordinates (values).

Graph: The graph of an inverse relation is the reflection of the original graph over the identity line, y=x. It may be necessary to restrict the domain on certain functions to guarantee that the inverse relation is also a function.

If the original function is a one-to-one function, the inverse will be a function. Use the horizontal line test to determine if a function is a one-to-one function.

True or False: The inverse of the graph shown below will be a function. Justify your answer.

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a and b pass the HLT so the inverse is a function. C fails the HLT so the inverse will not be a function.

Homework 5.5: Inverse Functions

1. Write the set of input-output pairs for the functions of f and g by filling in the blanks below. (The set F for the function f has been done for you.)

F={ (1 ,3 ) , (2 ,15 ) , (3 ,8 ) , (4 ,−2 ) , (5 ,9 ) }

G={ (−2 ,4 ) , ( , ) , ( , ) , ( , ) , ( , )}

2. Cindy thinks that the inverse of f ( x )=x−2 is g ( x )=2−x. To justify her answer, she calculates f (2 )=0 and then substitutes the output 0 into g to get

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g (0 )=2, which gives back the original input. Show that Cindy is incorrect by using other examples from the domain and range of f .

Find the inverse for these functions:

3. f ( x )=4 x+12 4. f ( x )= x+3x

5. True or False: Since f (x) is a reflection 6. Graph the inverse of the function below:

of g ( x ) , g(x ) is also the inverse of f (x).

Justify your answer.

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Lesson 5.6: Average Rate of Change

Learning Goal: In geometry you learned how to find the slope of a line over a given interval (2 points)

After today’s lesson you should be able to:

Find the average rate of change of a function over a given interval [a ,b] Apply average rate of change to word problems

Average Rate of Change:

The average rate of change of the function y=f (x ) between x=a∧x=b is

average rate of change= change∈ ychange∈x

=y2− y1x2−x1

=f (b )− f (a)

b−a

Warm up: Find the slope of the function over the given intervals

1. from x=1¿ x=4 2. from (−2 ,8 ) ¿(15 ,−22)

−11−14−1

=−123

=−4 −22−815−(−2)

=−3017

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3. from point A to point B riserun

=21=2

Discuss with a partner:

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1. Given the graph above, how do you find the average rate of change over the interval [3 ,6 ]? Find the slope of points (3 ,23 )∧(6 ,27)

2. Which interval has a larger average rate of change [3 ,6 ]∨[6 ,8 ]? Why?

[3 ,6 ] has a steeper slope!

The average rate of change is the slope of the secant line between 2 points of a graph. It will tell you if the graph is generally increasing or decreasing over a specific interval!

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Example 1: Given the function f ( x )=−x2+4 x−1, find the average rate of change from x=1¿ x=5. What does this tell you about the graph of the function over the interval [1,5]?

average rate of change= change∈ ychange∈x =

f (b )−f (a)b−a

f (5 )−f (1)5−1

(−(5 )2+4 (5 )−1)−(−(1 )2+4 (1 )−1)5−1

(−25+20−1)−(−1+4−1)4

−25+20−1+1−4+14

−84

−2 The graph is mostly decreasing between [1,5]

Example 2: Suppose an object is thrown upward with an initial velocity of 52 feet per second from a height of 125 feet. The height of the object t seconds after it is thrown is given by h (t )=−16 t2+52t +125

Find the average velocity in the first three seconds after the object is thrown.

How do you know this is asking for the average rate of change?

Average Velocity

First three seconds means: Interval = [0 ,3 ]

average rate of change= change∈ ychange∈x =

f (b )−f (a)b−a

h (3 )−h (0)3−0

¿¿

(−16 (9 )+156+125 )−(−16 (0 )+0+125)3

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(−144+156+125 )−(0+0+125)3

137−1253

123

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Example 3: An object travels such that its distance, d, away from its starting point is shown as a function of time, t, in seconds, in the graph below.

Is the average speed of this object greater on the interval 0≤ t ≤5∨11≤ t ≤14? Justify your answer. Means find slope!

0≤ t ≤5 11≤ t ≤14 11≤ t ≤14 is greater because

f (5 )−f (0 )5−0

f (14 )−f (11)14−11 the slope 5>4.

20−05

79−643

205

153

4 5

38

Example 4: Forrest Gump went running for several hours. The table below represents the number of miles he had run, m, after t hours. Is the average speed constant for this function? Justify your answer.

Compare the slopes over the intervals [2 ,4 ]∧[4 ,7 ]

18−104−2

=82=4∧30−18

7−4=123

= 4∧30−107−2

=205

=4

All slopes are the same so it is constant.

Extend your thinking: The table below represents a linear function. Fill in the missing entries.

Because it is a linear function the slope “average rate of change” will be constant

[1 ,5 ] f (5 )− f (1 )5−1

=1−(−5)5−1

=1+54

=64=32

[5 ,11 ] [11 , x ] [19,45]

f (11)−f (5 )11−5

=32 f ( x )−f 11¿ ¿

x−11=32

f (45 )−f (19)45−19 =

32

y−16

=32 f ( x )−f 11¿ ¿

x−11=32

y−2226

=32

18=2 y−2 22−10x−11

=32 78=2 y−44

20=2 y 12x−11

=32 122=2 y

10= y 24=3 x−33 61= y

39

57=3 x

19= x

Homework 5.6: Average Rate of Change

Find the average rate of change for the following functions over the given interval. What does the average rate of change tell you about the function on the interval?

1. From x=0¿ x=3

2. From x=−1¿ x=1 3. f ( x )=−√x−2+5 in the interval [3 ,11 ]

40

4. Which of the two functions has the largest average rate of change from x=−2¿ x=4?

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