function notation and making predictions section 2.3
TRANSCRIPT
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