Introduction to Functions Unit 4 Lesson 1

Objective: Define terms related to functions Determine if a relation is a function Evaluate a function

Definitions An ordered pair – used to locate points on a graph. (x, y) The first number, the x value corresponds to the horizontal axis The second number, the y value, corresponds to the vertical axis

Definition Relation – a set of ordered pairs

5 Ways to Represent a Relation Set of Ordered Pairs Mapping Table Graph Equation or function

Example Represent the following relation 4 ways: {(-2, -9), (-1, -5), (0, -1), (1, 3), (2, 7)} Mapping: X values Y values -2 -1 1 2 -9 -5 -1 3 7

{(-2, -9), (-1, -5), (0, -1), (1, 3), (2, 7)} Table: Graph: X Y -2 -9 1 3 2 7 Equation:

Definitions Domain – input values or x- values of a relation Range – output values or the y-values of a relation

Example {-4, -3, 0, 2, 3} Find the Range: {-4, -1, 0, 1, 2} Given the relation: {(-3, 2), (0,-4), (2,0), (-4, -1), (3, 1)} Find the domain: {-4, -3, 0, 2, 3} Find the Range: {-4, -1, 0, 1, 2}

Given the graph Find the domain: Find the range: {-4, -2, 2, 3} {-5, -4, 1, 5}

Definition A function is a relation such that for every x there is one and only one y (X can’t repeat)

Examples Determine if the following are functions. If not a function, explain why. {(0,2), (1,5), (6,3), (7,-5)} {(-3, 1), (-2, 6), (-1,0), (1, 0), (2, 6)} {(0, 2), (4,3), (0, -2)} Function Function Not a Function

Example Is this a function? (Not in notes) y 2 1 -1 2 Yes

Is this a function? x y -3 -1 4 -1 3 5 Yes

Is this a function? x y -7 -9 8 2 2 5 7 Yes

Is this a function? -3 -2 4 -5 8 10 12 No

Is this a function? -1 -2 -3 -4 3 4 5 No

Is this a function? X Y -2 -1 7 2 3 Yes

Is this a function X Y -2 -1 1 2 7 No

Another Definition The vertical line test – If a vertical line intersects the graph more than once, then the graph is not a function

Determine if the following are functions and give the range & domain of the each Yes (-∞, 5] (-∞, 2]

Determine if the following are functions and give the range & domain of the each Yes (-4, ∞) (-1, ∞)

Determine if the following are functions and give the range & domain of the each (-∞, 3] [-2, ∞)

Determine if the following are functions and give the range & domain of the each Yes (-∞, ∞) (-∞, ∞)

Determine if the following are functions and give the range & domain of the each [0, ∞) (-∞, ∞)

Determine if the following are functions and give the range & domain of the each Yes [-6, 6] [0, 6]

Determine if the following are functions and give the range & domain of the each [-4,4] [-4,4]

Determine if the following are functions and give the range & domain of the each [-5] (-2, 6)

Function Notation Read as f of x Does not mean multiply f times x

Evaluating a function To evaluate a function, replace the ‘x’ with the ‘number’ in the function, and then do the math.

Examples Find f(3) Every place there is an x, replace it with 3

Examples Find f(-1) Every place there is an x, replace it with -1

Examples Find f(a) Every place there is an x, replace it with a f(a) = 2(a) - 4 f(a) = 2a - 4

Examples Find f(a-3) Every place there is an x, replace it with a - 3 f(a=3) = 2(a-3) - 4 f(a - 3) = 2a - 6 – 4 f(a – 3) = 2a - 10

Example 2 Given: f(x) =2x2 + x -3 Evaluate f(-1) f(-1) = 2(-1)2 + (-1) -3 f(-1) = 2 - 1 – 3 = -2

Example 2 Given: f(x) =2x2 + x -3 Evaluate f(-3) f(-1) = 2(-3)2 + (-3) -3 f(-1) = 18 + (-3) – 3 = 12

Example 3 Replace the x with ‘a’ f(a) = f(x) =2a2 + a -3 Given: f(x) =2x2 + x -3 Find f(a) Replace the x with ‘a’ f(a) = f(x) =2a2 + a -3

Example 3 Replace the x with ‘a + 1’ f(a+1) = f(x) =2(a+1)2 + (a+1) -3 Given: f(x) =2x2 + x -3 Find f(a+1) Replace the x with ‘a + 1’ f(a+1) = f(x) =2(a+1)2 + (a+1) -3 Simplify. Remember to FOIL!      

Quick Review What is a relation? What is a domain? What is the range? What is a function How do you say this: f(x) What is the vertical line test