Module 3.3

Modeling With Functions

What is function notation, and how can youuse functions to model real-world situations?

P. 115

P. 127In the winter, more electricity is used when the outside temperature goes down, and less is used when the outside temperature rises.

The ___________________ depends on the ____________.

The dependent variable is ________________________. The independent variable is _________________.

How do you know? _____________________________________________________

The cost of shipping a package is based on its weight.

The ___________________ depends on the ____________.

The dependent variable is ________________________. The independent variable is _________________.

The faster Tom walks, the quicker he gets home.

The ___________________ depends on the ____________.

The dependent variable is ________________________. The independent variable is _________________.

Write an algebraic EXPRESSION to represent this situation: Amanda charges $10 per hour for babysitting.

10x where x is the number of hours.

Write an algebraic EQUATION using 2 variables to represent this situation.

𝒚 = 𝟏𝟎𝒙

The ______________________ is dependent on the _______________________.

The dependent variable is ____. The independent variable is ____.

This is an example of a function, because for every input (𝒙), there’s only one output (𝒚).

When we represent a function in FUNCTION NOTATION, we replace 𝒚 with 𝐟(𝐱), which is pronounced “f of x”.

So in FUNCTION NOTATION, that function/equation above is written as 𝒇(𝒙) = 𝟏𝟎𝒙, and pronounced “f of x equals 10x”.

Any variable will work for the independent variable. So the above could also be written as 𝒇(𝒉) = 𝟏𝟎𝒉, 𝒇(𝒕) = 𝟏𝟎𝒕, etc.

P. 128

P. 129For this situation:1) Identify the independent and dependent variables2) Write an equation in function notation3) Use the equation to solve the problem.

Function Notation

Once you’ve written a function rule,you can evaluate it for any value

of the independent variableby substituting that value into the rule

and simplifying.

You do this all the time in your head!

Function Notation

P. 129For this situation:1) Identify the independent and dependent variables2) Write an equation in function notation3) Use the equation to solve the problem.

Stan, a local delivery driver, is paid $3.50 per mile driven plus a daily amount of $75. On Monday, he’s assigned a route that’s 30 miles long. How much is he being paid for that day?

Kate earns $7.50 per hour. How much money will she earn after working 8 hours?

P. 129For these situations:1) Identify the independent and dependent variables2) Write an equation in function notation3) Use the equation to solve the problem.

The domain is all possible x-values or inputs.The range is all possible y-values or outputs.

Once you create a function to describe a real-world situation , you must choose a reasonable domain (x-values) and range (y-values). But – not all numbers are reasonable choices. For example, the length of an object can’t be negative, and only whole numbers can represent a number of people.

P. 130

Write a function in function notation for the situation, then find a reasonable domain and range.

Manuel has already sold $20 worth of tickets to the school play. He has 4 tickets left to sell at $2.50 per ticket. What is the total amount collected from ticket sales?

𝒚 = $𝟐. 𝟓𝟎 ∙ 𝒙 + $𝟐𝟎

or 𝒚 = 𝟐. 𝟓𝒙 + 𝟐𝟎

And in Function Notation: 𝒇(𝒙) = 𝟐. 𝟓𝒙 + 𝟐𝟎

How do you find all possible values of the range? Substitute all domain values into the function rule, one at a time, and evaluate.

When x = 0, it becomes 2.5(0) + 20 = 0 + 20 = 20. We write that as 𝒇 𝟎 = $𝟐𝟎When x = 1, it becomes 2.5(1) + 20 = 2.5 + 20 = 22.50. We write that as 𝒇 𝟏 = $𝟐𝟐. 𝟓𝟎When x = 2, it becomes 2.5(2) + 20 = 5 + 20 = 25. We write that as 𝒇 𝟐 = $𝟐𝟓. 𝟎𝟎When x = 3, it becomes 2.5(3) + 20 = 7.5 + 20 = 27.50. We write that as 𝒇 𝟑 = $𝟐𝟕. 𝟓𝟎When x = 4, it becomes 2.5(4) + 20 = 10 + 20 = 30. We write that as 𝒇 𝟒 = $𝟑𝟎. 𝟎𝟎

So the range is {$20, $22.50, $25, $27.50, $30}.

We said the domain is all possible x-values – OR INPUTS.So the input to the function doesn’t have to be an “x”; it could be any variable.Meaning, the function above could be written as 𝒇(𝒕) = 𝟐. 𝟓𝒕 + 𝟐𝟎, or 𝒇(𝒑) = 𝟐. 𝟓𝒑 + 𝟐𝟎, etc.

The domain is all possible x-values (or inputs). In this case, Manuel has only 4 tickets to sell, meaning possible values for x are 0, 1, 2, 3, and 4. So a reasonable domain is {0, 1, 2, 3, 4}.

P. 130Function Notation: 𝒇(𝒙) = 𝟐. 𝟓𝒙 + 𝟐𝟎

Don’t forget the zero!

Write a function in function notation for the situation, then find a reasonable domain and range.P. 130

A telephone company charges 25 cents per minute for the first 5 minutes of a call, plus a 45 centconnection fee per call. What is the total cost (in dollars) of making a call?

Let m represent the number of minutes used.

Or: 𝒇 𝒎 = 𝟎. 𝟐𝟓𝒎 + 𝟎. 𝟒𝟓

The charges only occur if a call is made, so a reasonable domain is { _____________________ }

Substitute these values into the function above to find the range.

And you get: { _____________________________________________________ }

Write a function in function notation for these situations, then find a reasonable domain and range.P. 130

The temperature early in the morning is 17 °C. The temperature increases by 2 °C for every hourfor the next 5 hours. What is the new temperature?

Flora earns $8.50 per hour proofreading advertisements at a local newspaper. She worksno more than 5 hours a day. What are her earnings?

How would you write 𝒙 + 𝒚 = 𝟔 in function notation?First solve for y: Subtract x from both sides, so you’d get 𝒚 = 𝟔 − 𝒙 or 𝒚 = −𝒙 + 𝟔,then replace y with f(x). So the answer would be: 𝒇(𝒙) = −𝒙 + 𝟔

How would you write 𝟑𝒙 + 𝟐𝒚 = 𝟔 in function notation?

A function can be thought of as a machine that processes inputs in a particular wayand always produces the same output for a given input.For example, when you input water into an ice machine, the output is always ice cubes. Can you think of other real-world objects that have an input and an output?

Create a function such that 𝒇(𝟑) = 𝟏𝟐, and describe a real-world situation that the function could represent.