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Functions and Variations

Functions and Variations. Functions and variations deal with relationships between a set of values of one variable and a set of values of other variables. Functions. Functions are very specific types of relations. Before defining a function, it is important to define a relation. Relations.

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Functions and Variations

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  1. Functions and Variations

  2. Functions and variations deal with relationships between a set of values of one variable and a set of values of other variables.

  3. Functions • Functions are very specific types of relations. Before defining a function, it is important to define a relation.

  4. Relations • Any set of ordered pairs is called a relation. Figure 10-1 shows a set of ordered pairs. • A = {(-1,1)(1,3)(2,2)(3,4)} (0,0) Origin Example 1

  5. Domain and Range • The set of all x’s is called the domain of the relation. The set of all y’s is called the range of the relation.

  6. Plotting Points Domain Range

  7. Defining a function • The relation in Example 1 has pairs of coordinates with unique first terms. When the x value of each pair of coordinates is different, the relation is called a function. A function is a relation in which each member of the domain is paired with exactly one element of the range.

  8. All functions are relations, but not all relations are functions. A good example of a functional relation can be seen in the linear equation y = x + 1. The domain and range of this function are both the set of real numbers, and the relation is a function because for any value of x there is a unique value of y.

  9. A graph of a linear equation y = x + 1

  10. Graphs of functions • Vertical line test y = x^2 y = sin(x) y = IxI

  11. Graphs of relationships that are not functions x = 2

  12. Determining domain, range, and if the relation is a function • B = {(-2, 3)(-1,4)(0,5)(1,-3)} • Domain: {-2, -1, 0, 1} • Range: {3, 4, 5, -3} • Function: yes

  13. Domain: {-2, -1, 1, 2} • Range: {-2, -1, 2} • Function: Yes or No?

  14. Domain: {2, 2, 3, 4, 5} • Range: {0, 1, 2, 3, 4} • Function: Yes or No?

  15. Finding the value of functions • The value of a function is really the value of the range of the relation. Given the function • f = {(1, -3)(2, 4)(-1, 5)(3, -2)} • The value of the function is 1 is -3, at 2 is 4, and so forth. This is written f(1) = -3 and f(2) = 4 and is usually read, “f of 1 = -3 and f of 2 = 4.” The lowercase letter f has been used here to indicate the concept of function, but any lowercase letter might have been used.

  16. Let h = {(3, 1)(2, 2)(1,-2)(-2, 3)} Find each of the following. • h(3) = • h(2) = • h(1) = • h(-2) =

  17. If g(x) = 2x + 1, find each of the following. • g(-1) = • g(2) =

  18. Variations • A variation is a relation between a set of values of one variable and a set of values of other variables.

  19. Direct Variation • In the equation y = mx + b, if m is a nonzero constant and b = 0, then you have the function y = mx (often written y = kx), which is called a direct variation. That is, you can say that y varies directly as x or y is directly proportional to x. In this function, m (or k) is called the constant of proportionality or the constant of variation. The graph of every direct variation passes through the origin.

  20. Graph y = 2x • Create a T chart. • (0,0) • (1,2) • (2,4)

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