# Capacities

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Slide 1

﻿Capacities Domain and Range

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Functions versus Relations A "relation" is only a relationship between sets of data. A "capacity" is a very much carried on connection, that is, given a beginning stage we know precisely where to go.

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Example People and their statures, i.e. the matching of names and statures. We can think about this connection as requested match: (stature, name) Or (name, tallness)

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Example (proceeded)

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Jim Kiki Rose Mike Joe Mike Rose Kiki Jim Both charts are relations (stature, name) is not all around carried on . Given a stature there may be a few names relating to that tallness. How would you know then where to go? For a connection to be a capacity, there must be precisely one y esteem that corresponds to a given x esteem .

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Conclusion and Definition Not each connection is a capacity. Each capacity is a connection. Definition: Let X and Y be two nonempty sets. A capacity from X into Y is a connection that partners with every component of X precisely one component of Y .

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Recall, the chart of (tallness, name): What happens at the stature = 5?

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Vertical-Line Test An arrangement of focuses in the xy-plane is the chart of a capacity if and just if each vertical line crosses the diagram in at most one point .

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Representations of Functions Verbally Numerically, i.e. by a table Visually, i.e. by a diagram Algebraically, i.e. by an unequivocal recipe

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Ones we have settled on the representation of a capacity, we ask the accompanying inquiry: What are the conceivable x-values (names of individuals from our illustration) and y-values (their comparing statures) for our capacity we can have?

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Recall, our case: the blending of names and statures. x=name and y=height We can have numerous names for our x-esteem, however shouldn't something be said about statures? For our y-values we ought not have 0 feet or 11 feet, since both are incomprehensible. In this way, our gathering of statures will be more prominent than 0 and less that 11.

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We ought to give a name to the gathering of conceivable x-values (names in our illustration) And To the accumulation of their relating y-values (statures). Everything must have a name 

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Y=f(x) (x, f(x)) x Variable x is called free factor Variable y is called subordinate variable For comfort, we utilize f(x) rather than y. The requested combine in new documentation gets to be: (x, y) = (x, f(x))

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Domain and Range Suppose, we are given a capacity from X into Y. Review, for every component x in X there is precisely one comparing component y=f(x) in Y. This component y=f(x) in Y we call the picture of x. The space of a capacity is the set X. That is an accumulation of all conceivable x-values. The scope of a capacity is the arrangement of all pictures as x fluctuates all through the area.

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Our Example Domain = {Joe, Mike, Rose, Kiki, Jim} Range = {6, 5.75, 5, 6.5}

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More Examples Consider the accompanying connection: Is this a capacity? What is space and range?

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Visualizing space of

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Visualizing scope of

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Domain = [0, ∞) Range = [0, ∞)

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More Functions Consider a natural capacity. Range of a circle: A(r) = r 2 What sort of capacity is this? How about we see what happens in the event that we diagram A(r).

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Graph of A(r) = r 2 A(r) r Is this a right representation of the capacity for the range of a circle??????? Insight: Is area of A(r) right?

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Closer take a gander at A(r) = r 2 Can a circle have r ≤ 0 ? NOOOOOOOOOOOOO Can a circle have territory equivalent to 0 ? NOOOOOOOOOOOOO

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Domain and Range of A(r) = r 2 Domain = (0, ∞) Range = (0, ∞)

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Just an idea… Mathematical models that depict genuine wonder must be as precise as could be expected under the circumstances. We utilize models to comprehend the wonder and maybe to make an expectations about future conduct. A decent model rearranges reality enough to allow scientific counts however is sufficiently exact to give significant conclusions. Keep in mind, models have constraints. At last, Mother Nature has the last say.