Relations and Functions Functions vs Relations A relation

Relations and Functions

Functions vs. Relations • A "relation" is just a relationship between sets of information. • A “function” is a well-behaved relation, that is, given a starting point we know exactly where to go.

Example • People and their heights, i. e. the pairing of names and heights. • We can think of this relation as ordered pair: • (name, height)

Example (continued) Name Height (cm) Joe 165 Mike 170 Rose 160 Kiki 175 Jim 162

A relation is a set of ordered pairs. The domain is the set of all x values in the relation domain = {-1, 0, 2, 4, 9} These are the x values written in a set from smallest to largest {(2, 3), (-1, 5), (4, -2), (9, 9), (0, -6)} These are the y values written in a set from smallest to largest range = {-6, -2, 3, 5, 9} The range is the set of all y values in the relation This is a relation

A relation assigns the x’s with y’s 1 2 3 2 4 5 6 8 10 Domain (set of all x’s) Range (set of all y’s) 4 This relation can be written {(1, 6), (2, 2), (3, 4), (4, 8), (5, 10)}

AAfunction fffromset set. AAto toset set. BBisisaaruleof ofcorrespondence thatassignsto toeachelement xxin inthe theset set. AA exactly one elementyyin inthe theset set. B. B. re a ’ s ed x l Al sign as 1 2 3 4 5 2 4 6 8 10 Set A is the domain This is a function ---it meets our conditions No xh tha as m ass n one ore ign y ed Set B is the range Must use all the x’s The x value can only be assigned to one y

Let’s look at another relation and decide if it is a function. The secondition says each x can have only one y, but it CAN be the same y as another x gets assigned to. re a ’ s ed x l Al sign as 1 2 3 4 5 2 4 6 8 10 Set A is the domain This is a function ---it meets our conditions No xh tha as m ass n one ore ign y ed Set B is the range Must use all the x’s The x value can only be assigned to one y

Is the relation shown below a function? NO Why not? ? ? 2 was assigned both 4 and 10 1 2 3 4 5 2 4 6 8 10

Check this relation out to determine if it is a function. It is not---3 did not get assigned to anything 1 2 3 4 5 Set A is the domain 2 4 6 8 10 Set B is the range This is not a Must use all the x’s function---it doesn’t assign each x with a y The x value can only be assigned to one y

Check this relation out to determine if it is a function. This is fine—(all y’s don’t need to be used). 1 2 3 4 5 2 4 6 8 10 Set A is the domain This is a function Set B is the range Must use all the x’s The x value can only be assigned to one y

We commonly call functions by letters. Because function starts with f, it is a commonly used letter to refer to functions. This means the right hand side is a function called f This means the right hand side has the variable x in it The left side DOES NOT MEAN f times x like brackets usually do. The left hand side of this equation is the function notation. It tells us two things. We called the function f and the variable in the function is x.

Remember---this tells you what is on the right hand side---it is not something you work. It says that the right hand side is the function f and it has x in it. So we have a function called f that has the variable x in it. Using function notation we could then ask the following: This means to find the function f and instead of having an x in it, put a 2 in it. So let’s take the Find f (2). function above and make brackets everywhere the x was and in its place, put in a 2. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction

To determine by the graph if an equation is a function, we have the vertical line test. If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function. This is a function This is NOT a function This is a function

Graph: x -2 -1 0 1 2 f (x) -8 -1 0 1 8 This means “inverse function” (2, 8) (8, 2) x -8 -1 Let’s take the 0 values we got out of the function and 1 put them into the 8 f -1(x) -2 -1 0 1 2 (-8, -2) (-2, -8) inverse function and plot them Is this a function? What will “undo” a cube? Yes A cube root

If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function. This is a one-to-one function This is NOT a one -to-one function

More Examples • Consider the following relation: • Is this a function? (Vertical line test) • What is domain and range?

Visualizing domain of

Visualizing range of

More Functions • Consider a familiar function. • Area of a circle: • A(r) = r 2 • What kind of function is this? • Let’s see what happens if we graph A(r).

Graph of A(r) = r 2 A(r) r • Is this a correct representation of the function for the area of a circle? ? ? ? • Hint: Is domain of A(r) correct?

Closer look at A(r) = r 2 • Can a circle have r ≤ 0 ? • NOOOOOOO • Can a circle have area equal to 0 ? • NOOOOOOO

Domain and Range of A(r) = r 2 • Domain: x > 0 Range: y > 0

Name the Domain and Range From a Graph The set of ordered pairs may be an infinite number of points as described by a graph. Domain: all real numbers Range: y ≥ 0

Domain: x Range: all real numbers