3.1, 3.2, and Preview 3.3

3.1 and 3.2 Check

Work with your group to become experts of these sections!

Section 3.1 Review

What are Functions?

Function Machines

• A function is "a rule that takes in inputs and gives specific outputs".

• A function machine is one way to represent this process! Let's say our rule is "I double whatever is put into me". If I put in 5, my output will be 10.

• We can write our rule (or function) as an equation. For this example, my equation would be y = 2x. x is always my input, y is always my output.

Multiple Choice

If I input 7 into the function y = x + 3, what is my output?

• y = 4

• y = 10

• x = 4

• x = 10

Is it a Function?

• There is a special rule about functions: "To be a function, each input (x) is paired with exactly one output (y)! This means that if we put 3 into our machine and get 6, EVERY time we put in 3, we get 6.

• Note: It's ok if multiple inputs give the same output: for instance, we can have a machine that just turns any number into "0".

• This example is a function!

Fill in the Blanks

Fill in the Blanks

Multiple Choice

Is the relation a function? Why.

• Yes, because the x-value 11 has two y-values pair with it.
• Yes, because each x-value has only one y-value paired with it.
• No, because the x-value 11 has two y-values pair with it.
• No, because each x-value has only one y-value paired with it.

Multiple Choice

Is the relation a function? 
{(4,5), (4,-7), (4,2), (4,0), (4,9)}

• Yes
• No

Multiple Choice

Is this a function?

• Yes

• No

Vertical Line Test

There is a very easy way to tell if a graph is a function or not; and it's called the vertical line test! If you can draw a vertical line at any place on the graph, and it passes through the graph twice, it is NOT a function! So the graph of a circle is NOT a function, but the v-shaped graph is!

Multiple Choice

Is this graph a function or not a function? 

• Function 
• Not a Function 

Multiple Choice

Is this graph a function or not a function? 

• Function
• Not a Function

Multiple Choice

Is this graph a function or not a function? 

• Function
• Not a Function

Multiple Choice

Is this graph a function or not a function? 

• Function
• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Domain and Range

The domain and range are two ways to describe all the possible points in a graph. The domain represents all the possible values of x a graph can be (in the example to the right, x can be anything from -5 to infinity) and the range represents all the possible values of y a graph can be (in the example, y can be anything from negative infinity to 5).

Writing Domain and Range

• To write the domain and range, there are two different formats:

• If your graph is a bunch of unconnected points, we just list off all the x values in order for the domain, and all the y values in order for the range. We put them in curly brackets as well: D = {1, 2, 3, 4} and R = {5, 6, 7, 8}

• If your graph is one connected line, you have to write the domain and range as compound inequalities (find the highest and lowest values for each variable, and set them up like in the example): 1 ≤ x ≤ 4 and 5 ≤ y ≤ 8

Multiple Choice

Identify the domain of the following relation:
Remember domain is the "x" values

• -6, 14, 2, 28
• (-6,14), (0,32), (2,38), (4,44)
• -6, 0, 2, 4
• 14, 32, 38, 44

Multiple Choice

For the function {(0,1), (1,-3), (2,-4), (-4,1)}, write the domain.
Remember:  Domain is x 
(x,y)

• D: {1, -3, -4,}
  R: {0, 1, 2, -4}
• D:{0, 1, 2, -4}
  R:{1, -3, -4}
• D:{0, 1, 2, 3, 4}
  R:{1, -3, -4}

Multiple Choice

What is the DOMAIN of this graph?

• -4 ≤ x ≤ 3

• -1 ≤ x ≤ 4

• 4 < x < 3

• -1 < x < 4

Section 3.2

Linearity

Linear or Nonlinear

• "Linear" is just another way we can describe functions. A linear function is one which can be represented by a straight line and a constant rate of change.

• To check the rate of change, see if your x values and y values go up by the same amount each time!

• Ex: If the pattern of my x values is that I "add 5" each time, and the y values I "minus 2" each time, it's linear.

Multiple Choice

Which graph is linear?

• Graph A
• Graph B
• Graph C
• Graph D

Multiple Choice

Is function represented by the table linear or nonlinear?

• linear
• nonlinear

Multiple Choice

Is function represented by the table linear or nonlinear?

• linear
• nonlinear

Linear Equations

• Instead of a graph or a set of points, you might be given an equation and asked if it's linear or nonlinear. Here is the rule: If you can simplify the equation to get all your y values on one side and your x values on the other, with no exponents and no variables in the denominator of a fraction, it's a linear equation.

• 2y + 3x = 15 is linear, because it's just one step away from having the x and y on different sides.

• 5xy + 3 = 7x is nonlinear, because there is no way to separate the x and y variables.

Multiple Choice

Is the following equation linear or nonlinear? 

• Linear 
• Nonlinear 

Multiple Choice

Is the following equation linear or nonlinear? 

• Linear 
• Nonlinear 

Multiple Choice

Is the following equation linear or nonlinear? 

• Linear 
• Nonlinear 

Multiple Choice

Is the following equation linear or nonlinear? 

• Linear 
• Nonlinear 

Multiple Choice

Is the following equation linear or nonlinear? 

• Linear 
• Nonlinear 

New: Continuous vs Discrete

• Last we left off, we were discussing continuous and discontinuous functions.

• Continuous means "connected". Any graph with all its points connected is continuous.

• Discrete means "disconnected". Any graph that is just a collection of points is discrete!

How do I Know from Words?

• We will need to know whether a situation is discrete or not just from word problems alone, with no graph to help us.

• For instance, here's a word problem: The linear function y = 1.5x represents the cost y (in dollars) of x bottles of orange juice. Each customer can buy a maximum of 5 bottles.

• Ask yourself: is it possible to buy half a bottle of orange juice here? Is it possible to buy any number of bottles that's not a whole number?

• If your x can only be counted in whole numbers (like bottles of orange juice), it is a discrete function. If it can be ANY number in a certain range, it is continuous.

Multiple Choice

How many cows are there in that field?

• continuous

• discrete

Multiple Choice

What is the temperature of this fire?

• continuous

• discrete

Multiple Choice

Discrete or continuous data?

• Discrete

• Continuous

Multiple Choice

Discrete or continuous data?

• Discrete

• Continuous

Multiple Choice

The time it takes for a light bulb to burn out. 

• Discrete
• Continuous

Multiple Choice

The number of suitcases lost by airlines. 

• Discrete
• Continuous

Multiple Choice

The number of orange Skittles in a bag. 

• Discrete
• Continuous