Coming after you with Ch. 2: Functions and Their Graphs

2.1: Functions

Learning Targets:

• Determine Whether a Relation Represents a Function

• Find the Value of a Function…plug it in, plug it in!

• Find the Domain if a Function…look at the x’s!

• Identify the Graph of a Function

• Obtain Information from the Graph of a Function

The set X is called the domain of the function.

For each element x in X, the corresponding element y in Y is called the image of x. The set of all images of the elements of the domain is called the range of the function.

Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates with each element of X a unique element of Y.

DOMAIN RANGE

X Y

f

x

x

x

y

y

Determine which of the following relations represent functions.

Not a function.

Function.

Function.

Not a function.(2,1) and (2,-9)both work.

Find the domain of the following functions:

A)

B)

With a rational expression be sure to consider all valuesthat may make it undefined.

C)

Even roots are real only for nonnegative numbers.

Theorem: Vertical Line Test

A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

x

y

Not a function.

x

y

Function.

4

0

-4(0, -3)

(2, 3)

(4, 0)(10, 0)

(1, 0)

x

y

Determine the domain, range, and intercepts of the following graph. Find f(1). How often does the line y=-1 intersect the graph? For what value does f(x)=3?

Finding values of a function

For the function defined by f(x) = x2 - 2x, evaluate:

(a) f(2) (b) f(x) + f(2) (c) f(x+2) (d) f(x+h)

(a) f(2) = 22-2(2) = 0 (b) f(x) + f(2) = x2 - 2x + 0

(c) f(x+2) = (x+2)2 - 2(x + 2) = x2 + 4x + 4 - 2x - 4

= x2 + 2x

(d) f(x+h) = (x + h)2 - 2(x + h) = x2 + 2xh + h2 - 2x - 2h