# 3.3 – Input-Output Machines

Objectives

• Picture functions as “input-output machines”
• Compare the values placed into a function with the notation returned from a function
• Identify domains and ranges of functions
• Map domains and ranges of functions with an input-output diagram
• Apply a rule to an input-output diagram and function notation

Key Terms

• Domain – The set of all possible input values of a function (x-values on a graph).
• Input – A number that is entered into a function.
• The x-value, input, independent variable, domain
• Input-Output Diagram – A diagram that relates the inputs of a function to its outputs.
• Input values are shown on the left, with arrows pointing to the output values on the right.
• The values on the left belong to the function’s domain.
• The values on the right belong to the function’s range.

• Output – The result of a function.
• The y-value, output, dependent variable, range
• Range – The set of all possible output values of a function (or y-values on a graph).

Notes

• Odd Number Function:  O(n) = 2n – 1
• Even Number Function:  E(n) = 2n

• A yard is equal in length to three feet.
• The function F(y) takes a measurement in yards (as input) and returns a measurement in feet (as output).
• So, the x-value (input) is actually listed as “y” because “y” stands for “yards”
• And, the y-value (output) is the answer in “feet”
• Function for Yards to Feet: F(y) = 3y
• Because, for every yard, you get 3 feet (so you have to multiply yards times 3 feet).

• Example: F(15.4) = ? ft
• Set up the function: F(15.4) = 3(15.4)
• Solve: F(15.4) = 46.2 ft
• 15.4 yards will produce 46.2 feet
• The result, in feet, of 15.4 yards is 46.2 feet

• S(n) is a function that squares numbers.
• Function: $S(n)=n^2$

• Example: What is the value of S(13)?
• Set up the function: $S(13)=13^2$
• Solve: $S(13)=169$
• The square of a 13 results in the value 169
• The result of squaring 13 is 169

• A(r) is a function that gives the area of a circle with radius r.
• Function: $A(r) = 3.14r^2$

• Example: What is the value of A(3)?
• Set up the function: $A(3) = 3.14\bullet3^2$
• Solve: $A(3) = 28.26$
• The area of a circle with a radius of 3 is 28.26 units squared.

• Example of the Odd Number Function
• Solve: O(n) = 2n – 1 for O(30)
• Step 1: Substitute 30 for n: O(30) = 2(30) – 1  =  60 – 1  =  59
• Step 2: Notice if this is an Odd or Even number. This is an odd number, so it is the 30th odd number.
• Example of the Even Number Function
• Solve:  E(n) = 2n for E(7)
• Step 1: Substitute 7 for n:  E(7) = 2(7) = 14
• Step 2: Notice if this an Odd or Even number.  This is an even number, so it is the 7th even number.