1. determine whether a relation is a function. 2. find the domain of functions. 3. evaluate...

1. Determine whether a relation is a function.

2. Find the domain of functions.

3. Evaluate piecewise-defined and greatest integer functions.

3.1 Functions

Describe the set of inputs, the set of outputs, and the rule for the following functions:

A. The cost of a tank of gasoline depends on the number of gallons purchased.

B. The volume of a cube depends on the length of its edges.

Example #1Determining Inputs and Outputs of Functions

Input: Number of gallons purchasedOutput: Cost of a gasRule: Price per gallon times number of gallons

Input: Length of edge of cubeOutput: Volume of cubeRule: V = s3

C. A hydrologist records the depth of a pond in a certain place over a 6 month period in the form of a graph. The graph shows the depth recorded each month.

Example #1Determining Inputs and Outputs of Functions

Input: MonthOutput:DepthRule: Month/Depth Graph

A.

B.

Example #2Determining Whether a Relation is a Function

Inputs 1 2 3 4

Outputs 8 7 8 9

Inputs 10 12 11 10

Outputs 8 7 8 8

Yes, every input has exactly one output, even though 1 & 3 share the same output it is still a function.

Yes, every input has exactly one output, even though 10 repeats as an input twice, the output is the same.

Find the indicated values of

A. g(1)

B. g(−3)

C. g(y+2)

Example #3Evaluating a Function

g(x) 2x 2 7

39712 2

525732 2

15827882

7442722

22

22

yyyy

yyy

Find each output for

1. First find

Example #4Finding a Difference Quotient

s(x) 3x 2 2x h 0

s(x h)

hxhxhx

hxhxhx

hxhx

22363

2223

23

22

22

2


2. Next find


s(x) 3x 2 2x h 0

s(x h) s(x)

hhxh

xxhxhxhx

xxhxhxhx

236

2322363

2322363

2

222

222


3. Finally find


s(x) 3x 2 2x h 0

s(x h) s(x)h

236

236 2

hxh

hhxh

A.

Example #5Determining if an Equation Defines a Function

2y 3 32x 11 0

Solve each equation for y, graph, and use the vertical line test to determine whether the line is a function.

3

3

3

2

1116

2

1116

11322

xy

xy

xy

It is a function, a vertical line only crosses once.

B.

Example #5Determining if an Equation Defines a Function

2x y 2 3 0

Solve each equation for y, graph, and use the vertical line test to determine whether the line is a function.

32

322

xy

xy

Even roots have two possible answers and require the ± symbol. Not a function, a vertical

line will cross the line twice.

The definition of domain is the set of all “input” values for which the function is defined. Defined is the key word here because any values that make the function undefined are excluded from the domain.

There are two situations that will make a function undefined:1. Dividing by 0.2. Taking an even root (for instance a square root) of a

negative number, which produces imaginary solutions.

To find the domain of a function, we must determine what value(s) of x, if any, will either make the denominator become zero OR make the radicand of an even root negative.

Domains of Functions

Find the domain for each function given.

A.

Example #6Finding Domains of Functions

f(x) 3x 4

If x is in the denominator, we must determine what values of x would make the denominator become zero, because 3/0 is undefined.

We do this by setting the denominator equal to zero and solving, remember, this value is excluded from the domain!

4

04

x

x Therefore the domain of this function is all reals except −4.

4,|

,44,

xxx


B.


4

5)(

2

xxf

First of all, notice that the denominator is NOT factorable because it has a +4 and not a -4.

ix

x

x

x

2

4

4

042

2

In this case, the only excluded values are imaginary, therefore the domain is the set of all reals.

xx |

,


C.


f(w) 1 3wThis time we have a radical, specifically a square root, so we must find the values of w which would make the radicand negative.

If the radicand cannot be negative, it must be greater than or equal to zero, because the square root of 0 is 0, and therefore defined.

3

1

13

031

w

w

w

3

1,|

3

1,

www

The domain is then all reals less than or equal to 1/3.


D.


3 32)( ttf

This time the root is a cube root and has no restrictions on its domain.

xx |

,

Robin cleans small passenger airplane interiors at an airport. At most, she will clean 24 planes per month. Robin pays a license fee of $150 per month to the airport. Her cost per plane (cleaning supplies, commute to airport, and salary of a helper) is approximately $82.50. She charges $229 to clean each plane. Let x represent the number of planes she cleans in a month.

A. Express Robin’s monthly revenue R as a function of x.

Example #7

xxR 229)(

Revenue is the gross amount of money made without taking expenses into consideration.


B. Express the monthly cost C as a function of x.

Example #7

15050.82)( xxC


C. Find the rule and the domain of the monthly profit function P.

Example #7

15050.146

15050.82229)(

)()()(

x

xxxP

xCxRxP

Since she can’t clean negative amounts of planes & at most 24 planes, the domain is [0, 24]

Given:

A. h(0)

B. h(3.5)

C. h(4)

Example #8Evaluating a Piecewise-Defined Function

3152

453)(

2

xifx

xifxxh

0 is in the domain of the second function because 0 < 3.

1515)0(2

3.5 does not belong to the domain of either function and is therefore undefined.

4 is in the domain of the first function because 4 ≥ 4.

435163543 2

For any integer x, round down to the nearest integer less than or equal to x.

Greatest Integer Function

xxf

xxf

)(

)(

OR

Sometimes this is called the floor function instead of the greatest integer function.

LetEvaluate:A. B. C. D. E.

F.

Example #9Evaluating the Greatest Integer Function

f( 209.99)f( 51)

xxf )(

)9.6(f)7(f

4

5f

10f

6

7

-210 (remember that -210 < -209)

-51

3 (the square root of 10 is approximately 3.2)

1 (5/4 is 1.25)

Don’t forget to round down.

1. determine whether a relation is a function. 2. find the domain of functions. 3. evaluate...

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