Function Notation and

Making Predictions

Section 2.3

Lehmann, Intermediate Algebra, 3edSection 2.3

• Rather than use an equation, table, graph, or words to refer to a function: name it• If f is the name use to represent y:

• We refer to “ ” as functional notation• We substitute for y in the equation

• does not mean f times x

Slide 2

IntroductionFunction Notation

f x

y f x Reads, “f of x”

f x

2 1y x 2 1f x x

f x

f x

Lehmann, Intermediate Algebra, 3edSection 2.3

• Substitute 4 for x in the equation :

• With function notation, the input is and output is • Substitute 4 for x into the equation :

• mean input leads to an output of

Slide 3

IntroductionFunction Notation

2 1: 2 4 1: 9y x y y

4x 9y

2 1y x

2 1f x x

9y 4x 4 9f

Lehmann, Intermediate Algebra, 3edSection 2.3

• Notice that is of the form

• The number is the value of y when x is 4• To find , we say that we evaluate the function

at

Evaluate

Slide 4

Introduction/Evaluating a FunctionFunction Notation

4 9f input outputf

4f

4f4x

Example 4 2 at 5.f x x

Lehmann, Intermediate Algebra, 3edSection 2.3

• Could have used “g” to name the function

• Common symbols are f, g, and h.

Slide 5

Evaluating a FunctionFunction Notation

Solution

4 2:y x 4 2g x x

Lehmann, Intermediate Algebra, 3edSection 2.3

Find given

Slide 6

Evaluating FunctionsFunction Notation

Example 2f 22 3 .f x x x

Solution

Lehmann, Intermediate Algebra, 3edSection 2.3

Find given

Slide 7

Evaluating FunctionsFunction Notation

Example 3g 4 2

.5 1x

g xx

Solution

Lehmann, Intermediate Algebra, 3edSection 2.3

Find given

Slide 8

Evaluating FunctionsFunction Notation

Example h a 3 5.h a x

Solution

Lehmann, Intermediate Algebra, 3edSection 2.3

Some input–output pairs of a function are shown in the table. Find and

Slide 9

Using a Table to Find an Output and an InputFunction Notation

Example

Solution

7g

• The input of leads to • So,• Inputs leading to are• So, for ,

7 12g 7x 12y

9.g x

9y 4 and 6x x 9g x 4 and 6x x

Lehmann, Intermediate Algebra, 3edSection 2.3

Let . Find

Slide 10

Using An Equation to Find an Output and an InputFunction Notation

Example

Solution

31

2f x x 4 and when 4.f x f x

Lehmann, Intermediate Algebra, 3edSection 2.3

• Substitute and solve for x:

Slide 11

Using An Equation to Find an Output and an InputFunction Notation

Solution Continued

34 for in 1

2f x f x x

Lehmann, Intermediate Algebra, 3edSection 2.3

• Verify solution using graphing calculator• Use table in Ask mode

Slide 12

Using An Equation to Find an Output and an InputFunction Notation

Graphing Calculator

Lehmann, Intermediate Algebra, 3ed

• Blue arrow shows that the input leads to output of • So,

Section 2.3

A graph of a function is sketched. Find

• refers to when• We want y when

Slide 13

Using a Graph to Find the Values of x and f(x)Function Notation

Example

Solution 4 .f

• 4x

4 3f

4f

4x 4x

4y

Lehmann, Intermediate Algebra, 3ed

• The line contains the point (0, 1)• So,

Section 2.3

A graph of a function is sketched. Find

• refers to when• We want y when

Slide 14

Using a Graph to Find the Values of x and f(x)Function Notation

Example

Solution 0 .f

• 0x

0 1f

0f

0x

Lehmann, Intermediate Algebra, 3ed

• Red arrow shows output originates from the input • So,

Section 2.3

A graph of a function is sketched. Find x when

• • Want is the value of x when

Slide 15

Using a Graph to Find the Values of x and f(x)Function Notation

Example

Solution 2.f x

• 2, so 2y f x y

2y 2y

6x 6x

Lehmann, Intermediate Algebra, 3ed

• Output originates from the input • So,

Section 2.3

A graph of a function is sketched. Find x when

• • Want is the value of x when

Slide 16

Using a Graph to Find the Values of x and f(x)Function Notation

Example

Solution 0.f x

• 0, so 0y f x y

0y 0y 2x

2x

Lehmann, Intermediate Algebra, 3edSection 2.3

The table shows the average salaries of professors at four-year colleges and universities.

Let s be the professors’ average salary(in thousands

Slide 17

Using an Equation of a Linear Model to Make Predictions

Using Function Notation with Models

Example

of dollars) at t years since 1900. A possible model is

1. Verify that the above function is the model.

1.71 113.12s t

Lehmann, Intermediate Algebra, 3edSection 2.3

• Graph the model and the scattergram in the same viewing window• Function seems to model the

data well

Slide 18

Using an Equation of a Linear Model to Make Predictions

Using Function Notation with Models

Solution

Example Continued2. Rewrite the equation with the

function name f. 1.71 113.12s t

Lehmann, Intermediate Algebra, 3edSection 2.3

• t is the independent variable• s is the dependent variable• f is the function name, so we rewrite • Substitute for s:

Slide 19

Using an Equation of a Linear Model to Make Predictions

Using Function Notation with Models

Solution

Example Continued3. Predict the average salary in 2011.

1.71 113.12f t t s f t

f t

Lehmann, Intermediate Algebra, 3edSection 2.3

• Represent the year 2011 by• Substitute 111 for t into

Slide 20

Using an Equation of a Linear Model to Make Predictions

Using Function Notation with Models

Solution

Example Continued4. Predict when the average salary will be $80,000.

1.71 113.12f t t 111t

Lehmann, Intermediate Algebra, 3edSection 2.3

• Represent average salary of $80,000 by • Since , substitute 80 for and solve for t

Slide 21

Using an Equation of a Linear Model to Make Predictions

Using Function Notation with Models

Solution

s f t80s

f t

Lehmann, Intermediate Algebra, 3edSection 2.3

• According to model, average salary will be $80,000 in • Using TRACE verify the predictions

Slide 22

Using an Equation of a Linear Model to Make Predictions

Using Function Notation with Models

Graphing Calculator

1900 113 2013

Lehmann, Intermediate Algebra, 3edSection 2.3

• When making a prediction about the dependent variable of a linear model, substitute a chosen value for the independent variable in the model. Then solve for the dependent variable.• When making a prediction about the independent

variable of a linear model, substitute a chosen value for the dependent variable in the model. Then solve for the independent variable.

Slide 23

Using an Equation of a Linear Model to make Predictions

Using Function Notation with Models

Summary

Lehmann, Intermediate Algebra, 3edSection 2.3

To find a linear model and make estimates and predictions,

1.Create a scattergram of data to determine whether there is a nonvertical line that comes close to the data points. If so, choose two points (not necessarily data points) that you can use to find the equation of a linear model.

2.Find an equation of your model.

Slide 24

Four-Step Modeling ProcessUsing Function Notation with Models

Process

Lehmann, Intermediate Algebra, 3edSection 2.3

3. Verify your equation by checking that the graph of your model contains the two chosen points and comes close to all of the data points.

4. Use the equation of your model to make estimates, make predictions, and draw conclusions.

Slide 25

Four-Step Modeling ProcessUsing Function Notation with Models

Process

Lehmann, Intermediate Algebra, 3edSection 2.3

In an example from Section 2.2, we found the equation . , where p is the percentage of

Slide 26

Using Function Notation; Finding InterceptsFinding Intercepts

Example

0.53 74.50p t

American adults who smoke and t years since 1990.

1. Rewrite the equation with the function name g.

0.53 74.50p t

Lehmann, Intermediate Algebra, 3edSection 2.3

• To use the name g, substitute for p:

2. Find . What does the result mean in this function?

•Substitute 110 for t in the equation

:

Slide 27

Using Function Notation; Finding InterceptsFinding Intercepts

Solution

0.53 74.50g t t

110g

g t

Example Continued

Solution

0.53 74.50g t t

Lehmann, Intermediate Algebra, 3edSection 2.3

•When t is 110, p is 16.2. According to the model, 16.2% of American adults will smoke in 2010.

3. Find the value of t when . What does is mean in this situation?

Slide 28

Using Function Notation; Finding InterceptsFinding Intercepts

Solution Continued

Example Continued 30g t

Lehmann, Intermediate Algebra, 3edSection 2.3

• Substitute 30 for in the equation and solve for t

•When t is 110, p is 16.2. According to the model, 16.2% of American adults will smoke

Slide 29

Using Function Notation; Finding InterceptsFinding Intercepts

Solution

Example Continued

g t

Lehmann, Intermediate Algebra, 3edSection 2.3

• The model estimates that 30% of Americans smoked in

• Verify work on graphing calculator table

Slide 30

Using Function Notation; Finding InterceptsFinding Intercepts

Solution Continued

Example Continued

1900 83.96 1984

4. Find the p-intercept of the model. What does it mean in this situation?

Lehmann, Intermediate Algebra, 3edSection 2.3

• Since the model is in slope-intercept form the p-intercept is (0, 74.50)• The model estimates that 74.5% of American adults

smoked in 1900• Research would show that this estimate is too high

model breakdown has occurred

Slide 31

Using Function Notation; Finding InterceptsFinding Intercepts

Solution

Example Continued

5. Find the t-intercept. What does it mean?

0.53 74.50g t t

Lehmann, Intermediate Algebra, 3edSection 2.3

• To find the t-intercept, we substitute 0 for and solve for t:

Slide 32

Using Function Notation; Finding InterceptsFinding Intercepts

Solution g t

Lehmann, Intermediate Algebra, 3edSection 2.3

• The t-intercept is (140.57, 0)• So, the model predicts that no Americans adults will

smoke in • Common sense suggest this probably won’t occur• Use TRACE to verify the p- and i-intercepts.

Slide 33

Using Function Notation; Finding InterceptsFinding Intercepts

Solution Continued

1900 140.57 2041

Lehmann, Intermediate Algebra, 3edSection 2.3

If a function of the form , where , is used to model a situation, then•The p-intercept is (0, b).•To find the coordinate of the t-intercept, substitute 0 for p in the model’s equation and solve for t.

Slide 34

Intercepts of ModelsFinding Intercepts

Property0m p mt b

Lehmann, Intermediate Algebra, 3edSection 2.3

Sales of bagged salads increased approximately linearly from $0.9 billion in 1996 to $2.7 billion in 2004. Predict in which year the sales will be $4 billion.

• Let s be the sales (in billions of dollars)• Let t be the years since 1990• We want an equation of the form

Slide 35

Making a PredictionUsing Data Described in Words to Make Predictions

Example

Solution

s mt b

Lehmann, Intermediate Algebra, 3edSection 2.3

• First find the slope

• Substitute 0.23 for m:• To find b we substitute 6 for t and 0.9 for s

Slide 36

Making a PredictionUsing Data Described in Words to Make Predictions

Solution Continued

2.7 0.90.23

14 6m

0.23s x b

Lehmann, Intermediate Algebra, 3edSection 2.3

• Then substitute –0.48 for b:

• To predict when the sales will be $4 billion, we substitute 4 for s in the equation and solve for t:

Slide 37

Making a PredictionUsing Data Described in Words to Make Predictions

Solution Continued

0.23 0.48s t

Lehmann, Intermediate Algebra, 3edSection 2.3

• The model predicts that sales will be $4 billion in

• Verify using a graphing calculator table

Slide 38

Making a PredictionUsing Data Described in Words to Make Predictions

Solution Continued

1990 19 2009

Lehmann, Intermediate Algebra, 3edSection 2.3

A store opens at 9 A.M. to 5 P.M., Mondays through Saturday. Let be an employee’s weekly income (in dollars) from working t hours each week at $10 per hour.

1. Find an equation of the model f.

•The employee’s weekly income (in dollars) is equal to the pay per hour times the number of hours worked per week:

Slide 39

Finding the Domain and Range of a FunctionDomain and Range of a Function

Example

I f t

Solution

10f t t

Lehmann, Intermediate Algebra, 3edSection 2.3

2. Find the domain and range of the model f.

• To find domain and range we consider input-output• Store is open 8 hours a day, 6 days a week, the

employee can work up to 48 hours per week• So, the domain is• Since hours worked is between 0 and 48 hours,

inclusively, the pay is between 0 and 48(10)

Slide 40

Finding the Domain and Range of a FunctionDomain and Range of a Function

Example Continued

Solution

0 48t

Lehmann, Intermediate Algebra, 3edSection 2.3

• Range is• The figures illustrate inputs

of 22, 35, and 48 being sent to the outputs 220, 350 and 480, respectively• Label the t-axis that

represents the domain and the part of the I-axis that represents the range

Slide 41

Finding the Domain and Range of a FunctionDomain and Range of a Function

Solution Continued 0 480f t