9.5 Functions

CORD MathMrs. SpitzFall 2006

Objectives

• Determine whether a given relation is a function, and

• Calculate functional values for a given function

Assignment

• Pgs. 377-378 #5-48 all

Ex. 1: Is {(5, -2), (3, 2), (4, -1), (-2, 2)} a function?

• Since each element of the domain is paired with exactly one element of the range, this relation is a function.

Ex. 2: Which mapping represents a function?

2

3

5

-1

4

3

0

2

1

XY

The mapping in this mapping represents a function since, for each element of the domain, there is only one corresponding element in the range.

Ex. 3: Which mapping represents a function?

3

4

6

-2

0

3

4

7

2

X Y The mapping in this relation does not represent a function since the element 4 in the domain maps to two elements, -3 and 4, in the range.

Ex. 3: Is the relation represented by the equation x + 2y = 8 a function?

• Substitute a value for x in the equation. What is the corresponding value of y? Is there more than one value for y? For example, if x is 2, then y is 3 and that is the only value of y that will satisfy the equation. If you try other values of x, you will see that there is always only one corresponding value of y. Therefore, the equation x + 2y = 8 represents a function.

Notes

• For equations like x + 2y = 8, it may not be easy to determine whether there is an element of the domain that is paired with more than one element of the range. Often it is simpler to look at the graph of the relation. Suppose you graph x + 2y = 8.

First solve for y

x + 2y = 8-x - x 2y = 8 – x 2 2 y = 8 – x 2

Make a table of values and graph the equation.

x y2 3

0 4

8 0

6

4

2

-2

5 10 15

x + 2y = 8

Now place your pencil at the left of the graph to represent a vertical line. Slowly move the pencil to the right across the graph.

For each value of x, this vertical line passes through no more than one point on the graph. This is true for EVERY function.

Vertical Line Test for a Function

• If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.

Ex. 4: Use the vertical line test to determine if each relation is a function.

Okay be careful. Note the point!

Functional Notation• Equations that represent functions can be

written in a form called functional notation. The equation y = 2x + 1 can be written in the form f(x) = 2x + 1. The symbol f(x) is read “f of x” and represents the value in the range of the function that corresponds to the value of x in the domain. For example f(3) is the element of the range that corresponds to the element x = 3 in the domain. We say f(3) is the functional value of f for x = 3.

Functional Notation

• Letters other than f are also used for names of functions.

• The ordered pair (3, f(3)) is a solution of the function f in the previous slide.

• You can determine a functional value by substituting the given value for x into the equation. For example, if f(x) = 2x +1 and x = 3, then f(3) = 2(3) + 1 or 7.

Ex. 5 If f(x) = 3x – 7, find each of the following:

A. Find f(2) f(2) = 3 (2) – 7 = 6 – 7 = -1

B. Find f(5) f(5) = 3 (5) – 7 = 15 – 7 = 8

B. Find f(-3) f(-3) = 3 (-3) – 7 = -9 – 7 = -16

Ex. 6 If g(x) = x2 – 2x + 1, find each of the following:

A. Find g(6a) g(6a) = (6a)2 – 2(6a) + 1

= 36a2 – 12a + 1

B. Find 6[g(a)] 6[g(a)] = 6[a2 – 2(a) + 1 = 6a2 – 12a + 6