3.1 Functions
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Transcript of 3.1 Functions
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 4Outputs 8 7 8 9
Inputs 10 12 11 10Outputs 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
744272222
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
Find each output for
2. Next find
Example #4Finding a Difference Quotient
s(x) 3x 2 2x h 0
s(x h) s(x)
hhxh
xxhxhxhx
xxhxhxhx
236
2322363
2322363
2
222
222
Find each output for
3. Finally find
Example #4Finding a Difference Quotient
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
21116
21116
11322
xy
xy
xy
It is a function, a vertical line only crosses once.
B.
Example #5Determining if an Equation Defines a Function
2x y2 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!
404
xx Therefore the domain of this function is
all reals except −4.
4,|
,44,
xxx
Find the domain for each function given.
B.
Example #6Finding Domains of Functions
45)( 2
x
xfFirst of all, notice that the denominator is NOT factorable because it has a +4 and not a -4.
ixx
x
x
24
4
042
2
In this case, the only excluded values are imaginary, therefore the domain is the set of all reals.
xx |,
Find the domain for each function given.
C.
Example #6Finding Domains of Functions
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.
31
13031
w
ww
31,|
31,
www
The domain is then all reals less than or equal to 1/3.
Find the domain for each function given.
D.
Example #6Finding Domains of Functions
3 32)( ttfThis 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.
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.
B. Express the monthly cost C as a function of x.
Example #7
15050.82)( xxC
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.
C. Find the rule and the domain of the monthly profit function P.
Example #7
15050.146
15050.82229)()()()(
xxxxP
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
3152453)(
2
xifxxifxxh
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
45f
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.