1 Section 2.1 Functions. 2 3 A relation is a correspondence between two sets. If x and y are two...
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Transcript of 1 Section 2.1 Functions. 2 3 A relation is a correspondence between two sets. If x and y are two...
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Section 2.1
Functions
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A relation is a correspondence between two sets.
If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that y depends on x, and we write x y.
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FUNCTION
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Determine whether each relation represents a function. If it is a function, state the domain and range.
{(-2, 3), (4, 1), (3, -2), (2, -1)}
{(2, 3), (4, 3), (3, 3), (2, -1)}
{(2, 3), (4, 1), (3, -2), (2, -1)}
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Determine if the equation defines y as a function of x.
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2y x
Determine if the equation defines y as a function of x.
22 1x y
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FUNCTION MACHINE
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2For the function defined by 3 2 , evaluate:f f x x x
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(a) For each x in the domain of f, there is exactly one image f(x) in the range; however, an element in the range can result from more than one x in the domain.
(b) f is the symbol that we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range.
(c) If y = f(x), then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x.
SummaryImportant Facts About Functions
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2
4(a)
2 3
xf x
x x
2(b) 9g x x
(c) 3 2h x x
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A rectangular garden has a perimeter of 100 feet. Express the area A of the garden as a function of the width w. Find the domain.
wA
A(w) = w(w-50)
Domain: 0 < w < 50
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2 3For the functions 2 3 4 1
find the following:
f x x g x x
424 23 xx
224 23 xx
31228 325 xxx
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2 3
4 1
x
x
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A) function
B) not a function
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A)
B)
C)
D)
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Section 2.2
The Graph of a Function
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Average Price of Gasoline in California, Adjusted for Inflation.
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Which of the following are graphs of functions?
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Which of the following are graphs of functions?
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Consider the function 1
xf x
x
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A) yes, a function
B) no, not a function
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A)
B)
C)
D)
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Section 2.3
Properties of Functions
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For an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.
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So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.
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Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.
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INCRE
ASIN
G
DECREASING
CONSTANT
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Where is the function increasing?
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Where is the function decreasing?
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Where is the function constant?
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3Use a graphing utility to graph 2 3 1 for 2 2.
Approximate where has any local maxima or local minima.
f x x x x
f
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3Use a graphing utility to graph 2 3 1 for 2 2.
Determine where is increasing and where it is decreasing.
f x x x x
f
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2Suppose that 2 4 3.g x x x
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A) even
B) odd
C) neither
Is the function shown below even, odd or neither?
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A) increasing
B) decreasing
C) constant
Is the function below increasing, decreasing or constant on the interval (0, 1)?
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Section 2.4
Library of Functions;
Piecewise-defined Functions
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The Square Root Function
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The Cube Root Function
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The Absolute Value Function
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WISE
FUN
CTIO
NS
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A) square root function
B) square function
C) cube function
D) cube root function
The graph shown below is a:
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Which is the correct graph for the function below?
A) B)
C) D)
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Section 2.5
Graphing Techniques;
Transformations
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A)
B)
C)
D)
What is the equation for the function show below?
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Which graph below is the graph of the function
A) B)
C) D)
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Section 2.6
Mathematical Models: Building Functions
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A)
B)
C)
D)
Bob wants to fence in a rectangular garden in his yard. He has 62 feet of fencing to work with and wants to use it all. If the garden is to be x feet wide, express the area of the garden as a function of x.
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Section 5.1
Composite Functions
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Suppose that an oil tanker is leaking oil and we want to be able to determine the area of the circular oil patch around the ship. It is determined that the oil is leaking from the tanker in such a way that the radius of the circular oil patch around the ship is increasing at a rate of 3 feet per minute.
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2 3Suppose that 2 3 4 1. Find:f x x g x x
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2 3Suppose that 2 3 4 1. Find:f x x g x x
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1Suppose that and 1f x g x x
x
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A)
B)
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D)
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A)
B)
C)
D)
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Section 5.2
One-to-One Functions;
Inverse Functions
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For each function, use the graph to determine whether the function is one-to-one.
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A function that is increasing on an interval I is a one-to-one function in I.
A function that is decreasing on an interval I is a one-to-one function on I.
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A) Yes
B) No
Is the function shown below a one-to-one function?
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Which of the following is the graph of the function below and its inverse?
A) B)
C) D)
