Relations & Functions Section 2-1

Definitions A relation is a description of the association between two sets of values. The set of input values is called the domain and the set of output values is called the range. For example, there is an association (hopefully!) between the color of a traffic light and the behavior of a driver approaching it.

Traffic Light Color Behavior Red Yellow Green Stop Slow down Speed up Maintain speed Behavior

School Schedule Period Class 1 2 3 4 5 6 Algebra 2 Gym Chemistry Study Geometry Class

Cell Phone Direction Pad Up Down Left Right Direction Calculator Inbox Pictures Ringtones Action

Multiplication 2 -1 0.5 -1.3 1 Number 4 -2 1 -2.6 2 x 2

School Schedule (again!) Math Science Gym Study Subject 1 2 3 4 5 6 Period

Functions A function is a relation in which each input value maps to exactly one output value. Which of the previous examples are functions?

Ordered Pairs When the input and output values are numbers, as in the multiplication example, we can think of the input and output values as x and y, and represent the relation as a collection of ordered pairs: {(2, 4), (-1, -2), (0.5, 1), (-1.3, -2.6), (0, 0), (1, 2)} We can also graph these points!

Ordered Pairs (cont’d) x y 2 4 -1 -2 0.5 1 -1.3 -2.6

Vertical Line Test When we graph a relation, we can use the vertical line test to determine whether or not it is a function. In order to be a function, the graph must have the property that any vertical line drawn through it only touches it once. This corresponds to each input (x) value having only one output (y) value.

Vertical Line Test (cont’d)

Here’s a problem… What if we wanted to expand the previous example to include more inputs and outputs, but following the same rule? We could write out some more ordered pairs: …(4, 8), (5, 10), (6, 12), (7, 14)… … but these are just a few! There are infinitely many possible ordered pairs that we could add to that relation.

… and a solution! We can represent the relation using the equation that describes the relationship between the inputs and outputs: y = 2x

Solution (cont’d) Now if want to know what output value is produced by the input value 27, we just plug 27 in for x: y = 2(27) y = 54 Similarly, if want to know what input value gives an output value of -13, plug -13 in for y: -13 = 2x x = -6.5

Example 1 A relation is defined by the equation: y = x2 + 3 What are some ordered pairs that are part of this relation? (-2, 7), (-1, 4), (0, 3), (1, 4), (2, 7), (3, 12) Is this relation a function? Yes! For each input (x), there is exactly one output (y) – to find it, just square and add 3!

No! The input value 5 has two different output values: 4 and -4 Example 2 A relation is defined by the equation: |y| = x – 1 What are some ordered pairs that are a part of the relation? {(4, 3), (8, -7), (5, 4), (5, -4)} Is this relation a function? No! The input value 5 has two different output values: 4 and -4

Domain and Range Recall that the domain is the set of input values, and the range is the set of output values. When a relation is given as an equation, the domain and range are often difficult to figure out. We need to think about all the possible values of x and y in the equation.

Example 1 y = x2 + 3 Given a number as input (x), is there always an output value for it? Yes – just square it and add 3. The domain of this relation is all real numbers.

Example 1 (cont’d) y = x2 + 3 Given a number as output (y), can we always find an input (x) to go with it? No! For example, try y = -1: -1 = x2 + 3 has no solution! In fact, we know that x2 ≥ 0 always, so the output, which is equal to x2 + 3, satisfies: x2 + 3 ≥ 3 The range of this relation is {y | y ≥ 3}

Example 2 |y| = x – 1 Given a number as input (x), is there always an output value for it? No! For example, try x = -5. |y|= -5 – 1 has no solution! In order to have a solution, we need: x – 1 ≥ 0, or solving, x ≥ 1 The domain of this relation is {x | x ≥ 1}

Example 2 (cont’d) |y| = x – 1 Given a number as output (y), can we always find an input (x) to go with it? Yes – we can always solve for x. The range of this relation is all real numbers.

Function Notation Recall that when a relation is a function, there is exactly one output value for each input value. With functions, we sometimes use function notation to represent the output value: f(x) = x2 + 3 Replaces y Function name Input value Rule for finding the output value

Function Notation (cont’d) f(x) = x2 + 3 f(x) is read “f of x” and refers to the output value when x is the input value. f(-5) refers to the output value when 5 is used as an input value. f(-5) = (-5)2 + 3 = 28 f(-5) does not mean to multiply the variable f by the number -5!

More examples f(x) = x2 + 3 f(a) = a2 + 3 f(2a) = (2a)2 + 3 = 4a2 + 3 f(b + 5) = (b + 5)2 + 3 = (b+5)(b+5) + 3 = b2 + 10b + 25 + 3 = b2 + 10b + 28