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Functions domain-range
- 1. FunctionsFunctions Domain and RangeDomain and Range
- 2. Functions vs. Relations • A "relation" is just a relationship between sets of information. • A “function” is a well-behaved relation, that is, given a starting point we know exactly where to go.
- 3. Example • People and their heights, i.e. the pairing of names and heights. • We can think of this relation as ordered pair: • (height, name) • Or • (name, height)
- 4. Example (continued) Name Height Joe=1 6’=6 Mike=2 5’9”=5.75 Rose=3 5’=5 Kiki=4 5’=5 Jim=5 6’6”=6.5
- 5. MikeJoe Rose Kiki Jim Joe Mike Rose Kiki Jim • Both graphs are relations • (height, name) is not well-behaved . • Given a height there might be several names corresponding to that height. • How do you know then where to go? • For a relation to be a function, there must be exactly one y value that corresponds to a given x value.
- 6. Conclusion and Definition • Not every relation is a function. • Every function is a relation. • Definition: Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X exactly one element of Y.
- 7. • Recall, the graph of (height, name): What happens at the height = 5?
- 8. • A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point. Vertical-Line Test
- 9. Representations of Functions • Verbally • Numerically, i.e. by a table • Visually, i.e. by a graph • Algebraically, i.e. by an explicit formula
- 10. • Ones we have decided on the representation of a function, we ask the following question: • What are the possible x-values (names of people from our example) and y-values (their corresponding heights) for our function we can have?
- 11. • Recall, our example: the pairing of names and heights. • x=name and y=height • We can have many names for our x-value, but what about heights? • For our y-values we should not have 0 feet or 11 feet, since both are impossible. • Thus, our collection of heights will be greater than 0 and less that 11.
- 12. • We should give a name to the collection of possible x-values (names in our example) • And • To the collection of their corresponding y-values (heights). • Everything must have a name
- 13. • Variable x is called independent variable • Variable y is called dependent variable • For convenience, we use f(x) instead of y. • The ordered pair in new notation becomes: • (x, y) = (x, f(x)) Y=f(x) x (x, f(x))
- 14. Domain and Range • Suppose, we are given a function from X into Y. • Recall, for each element x in X there is exactly one corresponding element y=f(x) in Y. • This element y=f(x) in Y we call the image of x. • The domain of a function is the set X. That is a collection of all possible x-values. • The range of a function is the set of all images as x varies throughout the domain.
- 15. Our Example • Domain = {Joe, Mike, Rose, Kiki, Jim} • Range = {6, 5.75, 5, 6.5}
- 16. More Examples • Consider the following relation: • Is this a function? • What is domain and range?
- 19. • Domain = [0, ∞) Range = [0, ∞)
- 20. More Functions • Consider a familiar function. • Area of a circle: • A(r) = πr2 • What kind of function is this? • Let’s see what happens if we graph A(r).
- 21. A(r) r • Is this a correct representation of the function for the area of a circle??????? • Hint: Is domain of A(r) correct? Graph of A(r) = πr2
- 22. Closer look at A(r) = πr2 • Can a circle have r ≤ 0 ? • NOOOOOOOOOOOOO • Can a circle have area equal to 0 ? • NOOOOOOOOOOOOO
- 23. • Domain = (0, ∞) Range = (0, ∞) Domain and Range of A(r) = πr2
- 24. Just a thought… • Mathematical models that describe real-world phenomenon must be as accurate as possible. • We use models to understand the phenomenon and perhaps to make a predictions about future behavior. • A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. • Remember, models have limitations. In the end, Mother Nature has the final say.