Advanced Algebra w/Trig

Chapter 2.1 Relations & Formulas

Target Goals:1. Identify the domain and range of relations and functions;

express the domain and range in set notation and interval notation

2. Identify if a relation is a function3. Evaluate functions

NEW VOCABULARY!1. A _______________ is a set of ordered pairs.2. The _______________ of a relation is the set of all

the first coordinates (x-coordinates) from the ordered pairs.

3. The _______________ is the set of all the second coordinates (y-coordinates) from the ordered pairs.

4. In a coordinate plane, the x- and y-axes meet at the origin, (0, 0), and divide the plane into four ___________________.

RELATIONDOMAIN

RANGE

QUADRANTS

NEW VOCABULARY!5. A _______________ illustrates how each

element of the domain is paired with an element in the range.

6. A _______________ is a relation in which each element of the domain is paired with exactly one element of the range.

MAPPING

FUNCTION

How can you tell if it’s a function?

• In a function, an element of the range can be paired with more than one element of the domain. But an element of the domain cannot be paired with more than one element of the range.

• When looking at ordered pairs, if you see the same y-value in two ordered pairs, it’s A FUNCTION. If you see the same x-value in two ordered pairs, but they have different y-values, it’s NOT A FUNCTION.

• So… if you don’t see any x-values being repeated, your good to go! It’s a function!!!

Ex 1) State the domain and range of the relation. Express the domain and range in set notation. Then determine whether the relation is a function.

{(-4, -2), (-3, 1), (0, -2), (1, 2), (3, 3)}

Domain: ________________

Range: __________________

Function: _____________

{-4, -3, 0, 1, 3}

{-2, 1, 2, 3}

YES!

Do you see any x-values repeated in more than one ordered pair? - If so, check the y-values to see if they are the same or different. Same = function. Different = not a function.- If not, it’s a function!

Ex 2) Determine whether the relation is a function.

-3-2-101

02468

Domain Range

(-3, 0)(-2, 2)(-1, 6)(0, 4)(1, 6)

Yes it’s a function!

Ex 3) State the domain and range of the relation. Express the domain and range in set notation. Then determine whether the relation is a function.

x

y

What are all of the ordered pairs?

(-1, 2), (0, -1), (2, -2), (3, 1), (3, -1)

DOMAIN: {-1, 0, 2, 3}

RANGE: {-2, -1, 1, 2}

FUNCTION?:

NO!

Relations: Discrete vs. Continuous

• DISCRETE RELATION– A relation in which the domain is a set of individual

points (like examples 1, 2, 3). – Use set notation for naming the domain and range.

• CONTINUOUS RELATION– When the domain of a relation has an infinite number

of elements and it can be graphed with a line or a smooth curve.

– Use interval notation for naming the domain and range.

Vertical Line Test• A way to test if a curve or graph is a function

or not.• If no vertical line intersects a graph in more

than one point, the graph represents a function.

• If a vertical line intersects a graph in two or more points, the graph does not represent a function.

Vertical Line Test

Ex 4) Graph y = 2x + 1, and determine the domain and range. Express the domain and range in interval notation. Then determine whether the equation is a function. State whether it is discrete or continuous.

x y 0 11 32 5-1 -1

Domain: (-∞,∞)Range: (-∞,∞)

Function? YESContinuous or Discrete?

Ex 5) Graph y = x2 + 1, and determine the domain and range. Express the domain and range in interval notation. Then determine whether the equation is a function. State whether it is discrete or continuous.

x y0 11 22 53 10-1 2-2 5-3 10

Domain: (-∞,∞)

Range: [1,∞)

Function: YES!!!

Discrete or Continuous?

Functions and Graphing

• When an equation represents a function, the variable (often x) with values making up the domain is called the INDEPENDENT VARIABLE.

• The other variable (often y) is called the DEPENDENT VARIABLE.

• The independent variable is graphed on the horizontal/x-axis.

• The dependent variable is graphed on the vertical/y-axis.

FUNCTION NOTATIONEquation• y = 2x + 1• Read: “y equals 2 times x

plus 1”

Function• f(x) = 2x + 1• Read: “f of x equals 2 times

x plus 1”

THESE MEAN THE SAME THING! f(x) = y

f(x) replaces the dependent variable.

Function’s name is f.

Independent variable used in the function is x.

Ex 6) Given f(x) = x3 – 3, find each value:

(a) f (-2)

f (-2) = (-2)3 – 3f (-2) = -8 – 3f (-2) = -11

(b) f (2t)

f (2t) = (2t)3 – 3f (2t) = 8t3 – 3