Unit 3: An Introduction to Functions Do Now: Can you think of real world situations where you would input something into a machine and get a different output? For example, a toaster is a machine. When bread is input in the machine the output is toast. In the table below list at least two machines along with their inputs and outputs. Input Machine Output Bread toaster toast   ATM Debit Card Cash Vending Machine Money M&Ms

A function is a rule that assigns each input to exactly one output A function is a rule that assigns each input to exactly one output. In the do now example, the machine is a function. You can think of a function as being a box with a special rule, where the input is the stuff that goes into the box and the output is the stuff that comes out of the box.

Example 1: Movie Title Box Put a movie title into the box and the output is the first letter of a movie title. (Only movie titles can go in.) Groundhog Day G 101 Dalmations ???

The stuff that goes IN the box (the INPUT) is called the DOMAIN. Domain and Range The stuff that goes IN the box (the INPUT) is called the DOMAIN. The stuff that spits OUT of the box (the OUTPUT) is called the RANGE. For the previous example, the domain is all movie titles that start with a letter (101 Dalmations is not in the domain.) And the range is all letters of the alphabet. (there is a movie title for each letter of the alphabet!)

Example 2: Now let’s try a box with a rule of adding 3 Example 2: Now let’s try a box with a rule of adding 3. Draw a diagram and choose several values to put into the box to complete the table. State the domain and range. Domain: all real numbers Range: all real numbers

Example 3: Create a table of input and output values for the function rule 3x – 5. Then state the domain and range of the function. Input Output -5 Domain = Range = All real numbers 1 -2 -4 -17 .5 -3.5

Example 3: Create a table of input and output values for the function rule 1 𝑥 . Then state the domain and range of the function. Input Output 1 Domain = Range = All real numbers except 0 UNDEFINED

Example 4: Create a table of input and output values for the function rule 1 𝑥−2 . Then state the domain and range of the function. Input Output 1 -1 Domain = Range = All real numbers except 2 2 UNDEFINED

Different Representations of Relations and Functions A. Mapping Diagrams: 1.) 2.) Function Yes No Function Yes No Domain Domain Range Range   {-2,0,1,3} {-2,2,4,5,6} {4,16,25,36} {-3,1,3,4}

B. Ordered pairs: 1.) {(-3, 2) (-1, 5) (2, 9) (-3, 0) 2.) {(1, 3) (2, 7) (6, 11) (21, 53)}   Function Yes No Function Yes No Domain Domain Range Range {-3,-1, 2} {1,2,6,21) {0, 2, 5, 9} {3,7,11,53}

Lesson 1: Day 2 Using Function Notation and Evaluating Functions The function f acts upon the input, x, which we put within parentheses. The output is shown on the other side of the = symbol. For example, f(bread)= toast (where the function is the toaster) f(Frozen) = F f(Bourne Identity) = B

To describe a function that can have an unlimited number of inputs, we must describe f using an equation. We use the letter x to represent the inputs (also the domain) This means the letter x is said to be the independent variable. The output (also the range) is typically represented by the letter y and is said to be the dependent variable (simply because its value depends on the value of the independent variable).

Using the rule “add 5”, let x be the input. Then: f(x) = x + 5 x goes in, x + 5 comes out. f(x) is the official output name

Now let’s do some examples using function notation. a.) If the input is 4, what is the output? f(4) = 4 + 5 f(4) = 9 b.) Find f(-2). f(-2) = -2 + 5 f(-2) = 3

Guided Practice Exercises: For each of the problems below, a) Write a function using function notation to model the description. b) Evaluate the following for each: f(0), f(-3) and f(7). c) State the domain and range of the function. 1.) Three times a number plus 8. a.) f(x) = 3x + 8 c.) domain = all real numbers range = all real numbers b.) f(0) = 8 f(-3) = -1 f(7) = 29

2.) The square of a number minus 4. a.) f(x) = x2 - 4 b.) f(0) = -4 f(-3) = 5 f(7) = 45 c.) domain = all real numbers range = all real numbers 3.) Twice a number subtracted from 36. a.) f(x) = 36 - 2x b.) f(0) = 36 f(-3) = 42 f(7) = 22 c.) domain = all real numbers range = all real numbers

4.) Six more than four times a number. b.) f(0) = 6 f(-3) = -6 f(7) = 34 c.) domain = all real numbers range = all real numbers a.) f(x) = 4x + 6 5.) Thirteen less than a number. a.) f(x) = x - 13 b.) f(0) = -13 f(-3) = -16 f(7) = -6 c.) domain = all real numbers range = all real numbers

Real World applications of Domain and Range 6.) Does the relationship represent a function? Explain. A. {students in a school} → {locker}   B. {telephone number } → {students in a school} C. {cars in a parking lot } → {license plate number} D. {U.S citizen who works} → {social security #} E. {Year} → {Total number of North Atlantic hurricanes} Yes No Yes Yes Yes Complete Function Tables Worksheet

Sometimes functions have some meaning and a name for them is chosen to reflect that meaning. For example, the temperature of a heated object varies depending on how long it has been removed from the heat source. Suppose T is the name of the function that describes the Temperature of a hamburger that is cooling. We use the lowercase t, standing for time, as the independent variable since the Temperature, T, of the hamburger depends on the time, t.

Suppose the Temperature function is shown below. Time (t) T Temperature (T) The Temperature function tells us that the moment the hamburger leaves the grill, it is 170° F. After 1 minute, the hamburger is 169° F. After 2 minutes, the hamburger is 167° F and so forth. 0 minutes 170° F 1 minute 169°F 2 minutes 167°F 3 minutes 165°F

a.) What is the value of T(3 minutes)? b.) Suppose the Temperature function is given by the equation T(t) = 170 – 3t where t varies from 0 to 25 minutes. What is the value of T(0), T(1), and T(10)? c.) What is the value of T at t = 0, 1, and 2 if the equation of T is given by T(t) = 3t + 20? 165˚F T(0) = 170˚F T(1) = 167˚F T(10) = 140˚F T(0) = 20˚F T(1) = 23˚F T(2) = 26˚F

Homework: Complete problem set in packet