functions objective: to determine whether relations are functions

Functions

Objective: To determine whether relations are functions.

Introduction to Functions• Definition: A function f from a set A to a set B is a

relation that assigns to each element x in set A exactly one element y in set B. The set A (or x-values) is the domain (or set of inputs) of the function f, and the set B (or y-values) contains the range (or set of outputs).

Introduction to Functions• Definition: A function f from a set A to a set B is a

relation that assigns to each element x in set A exactly one element y in set B. The set A (or x-values) is the domain (or set of inputs) of the function f, and the set B (or y-values) contains the range (or set of outputs).

• Another way to describe a function is: for every value for x there is one and only one value for y.

Characteristics of a Function

1. Each element in A must be matched with an element in B.



2. Some elements in B may not be matched with any element in A.




3. Two or more elements in A may be matched with the same element in B.




3. Two or more elements in A may be matched with the same element in B.

4. An element in A (the domain) cannot be matched with two different elements in B.

Rule of Four1. Verbally by a sentence that describes how the

input variable is related to the output variable.2. Numerically by a table or list of ordered pairs that

matches input values with output values.3. Graphically by points on a graph in a coordinate

plane in which the input values are represented by the horizontal (x) axis and the output values are represented by the vertical (y) axis.

4. Algebraically by an equation in two variables.

Example 1• Determine whether the relation represents y as a

function of x.a. The input value x is the number of representatives

from a state, and the output value, y is the number of senators.


function of x.a. The input value x is the number of representatives

from a state, and the output value, y is the number of senators.

Answer: This is a function. No matter what the input value is, the output is 2. This is called a constant function and its graph is a horizontal line.


function of x.b. Input, x Output, y 2 11 2 10 3 8 4 5 5 1

Example 1• Determine whether the relation represents y as a function

of x.b. Input, x Output, y 11 2 10 3 8 4 5 5 1Answer: This is not a function. For one x value, 2, there is

more than one y value, 11 and 10.


function of x.c.


function of x.c.

Answer: This is a function. For each x, there is only one y. We say that this graph passes the vertical line test, therefore it is a function.

Example 2• Which of the equations represents y as a function of

x?

a.

12 yx

Example 2• Which of the equations represents y as a function of x?

a.

To determine whether y is a function of x, try to solve for y in terms of x.

This is a function since for every x there is only one y.

12 yx

21 xy

Example 2• Which of the equations represents y as a function of

x?

b.

This is not a function, since for every value of x there are two ( + ) values for y.

12 yx1

12

xy

xy

Function Notation• An equation is of the form . A function is of

the form . This is called function notation. The input values are called x and the output values f(x). The expression f(2) means go to the function and replace x with a 2. In this example,

21 xy 21)( xxf

0)1(1)1(

1)0(1)0(

3)2(1)2(

2

2

2

f

f

f

Example 3• Let• find• g(2)• g(t)• g(x+2)

14)( 2 xxxg

5184)44(

1)2(4)2()2(

14)(

51)2(4)2()2(

22

2

2

2

xxxx

xxxg

tttg

g

You Try• Let• find• f(2)• f(-1)• f(x-2)

12)( 2 xxxf

You Try• Let• find• f(2)• f(-1)• f(x-2)

12)( 2 xxxf

961)2(2)2()2(

41)1(2)1()1(

11)2(2)2()2(

22

2

2

xxxxxf

f

f

Piecewise-Defined Functions• A function defined by two or more equations over a

specific domain is called a piecewise-defined function.

Example 4• Evaluate the function when x = -1, 0, and 1.

11

)(2

xx

xf00

xx


• When x = -1, we use the top equation.• When x = 0, we use the bottom equation.• When x = 1, we use the bottom equation.

11

)(2

xx

xf00

xx


• When x = -1, we use the top equation.• When x = 0, we use the bottom equation.• When x = 1, we use the bottom equation.

11

)(2

xx

xf00

xx

011)1(110)0(

21)1()1( 2

fff

The Domain of a Function• The domain of a function describes what values we

can put in the equation for x. There are a few things we look at:

1. A zero in the denominator of a fraction.2. A negative value in a square root.3. A value that doesn’t make sense to the problem.

This is called a physical restraint.

Example 5• Find the domain of each function.a. f {(-3, 0), (-1, 4), (0, 2), (2, 2), (4, -1)}

Example 5• Find the domain of each function.a. f {(-3, 0), (-1, 4), (0, 2), (2, 2), (4, -1)}

Answer: The domain are the values that we use for x. The domain is { -3, -1, 0, 2, 4 }

Example 5• Find the domain of each function.

b.51)(

x

xg


b.

• This domain of this function is all values that don‘t make the denominator zero, which occurs at -5. The domain is .

51)(

x

xg

5x

Example 5

• Find the domain of each function.

c. Volume of a sphere 3

34 rV

Example 5

• Find the domain of each function.

c. Volume of a sphere

Since r represents the radius of the sphere, this value must be positive. We can’t have a negative value. The domain is r > 0.

3

34 rV


d. 24)( xxh


d.

We need . We plot the zeros and do sign analysis.

_______|________|_________

24)( xxh

04 2 x

22


d.

We need . We plot the zeros and do sign analysis. The answer is [-2, 2].

_______|________|_________

24)( xxh

04 2 x

22

Class work

• Pages 197-198• 4, 6, 10, 14, 20, 26, 58, 66

Homework

• Page 197• 1-9 odd• Page 198• 13, 15, 17, 25, 27, 31, 35, 39• Page 199• 57, 59, 61

functions objective: to determine whether relations are functions

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