Understanding Functions and Relations in Math: Definitions, Examples, and Operations

1. SECTION 1.1 • FUNCTIONS

2. DEFINITION OF A RELATION 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, we say that x corresponds to y or that y depends upon x. The correspondence can be written as an ordered pair (x,y).

3. DEFINITION OF A RELATION Thus, a relation is simply a set of ordered pairs or a table which relates x and y values.

4. DEFINITION OF FUNCTION Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates with each element of X a unique element of Y. This is a special type of relation. For every x, there is only one y!

5. DEFINITION OF DOMAIN DEFINITION OF RANGE DOMAIN AND RANGE The set of all x values. The set of all y values. Also called “functional values”.

6. THE FUNCTION AS A “MAPPING” x-values y-values 8 1 Ordered Pairs: (1 , 2) 4 2 (4 , 8) 7 0 (7, - 3) -2 -3 (- 2, 0) DOMAIN RANGE

7. THE FUNCTION AS A “MAPPING” Consider 3 students whose names are mapped to their letter grades on the last History exam: For each person in the domain, there can be only one associated letter grade in the range. Jill A Frank B Sue C

8. THE SQUARING FUNCTION -2 Each element in the domain maps to its square. 0 -1 1 0 1 4 2 9 3

9. COUNTER-EXAMPLE: Ordered Pairs: (4, 1) (4, 2) (5, 3) 1 4 2 5 3 This is an example of a relation but not a function.

10. THREE WAYS TO REPRESENT A FUNCTION • NUMERICALLY - ordered pairs • SYMBOLICALLY - equation • GRAPHICALLY - picture

11. EXAMPLE Determine whether the relation represents a function: (a) {(1,4),(2,5),(3,6),(4,7)} YES!

12. EXAMPLE Determine whether the relation represents a function: (a) {(1,4),(2,4),(3,5),(6,10)} YES!

13. EXAMPLE Determine whether the relation represents a function: (a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)}

14. EXAMPLE Determine whether the relation represents a function: (a) {(- 3,9),(- 2,4),(0,0),(1,1),(- 3,8)} NO!

15. EVALUATING A FUNCTION AT A GIVEN X-VALUE f(x) = f(0) = f(2) = f(-2) = f(9) = x f(x) x 2 0 0 2 4 - 2 4 9 81 0 4 4 81

16. Symbolically, the squaring function can be represented as y = x 2 “FUNCTIONAL NOTATION” f(x) = x 2 Read: “f of x equals x squared”

17. EVALUATING A FUNCTION AT A GIVEN X-VALUE • For f(x) = 2x2 – 3x, find the values of the following: • (a) f(3) (b) f(x) + f(3) (c) f(-x) • - f(x) (e) f(x + 3) • (f)

18. FINDING VALUES OF A FUNCTION ON A CALCULATOR DO EXAMPLE 7

19. IMPLICIT FORM OF A FUNCTION When a function is defined by an equation in x and y, we say that the function is given implicitly. If it is possible to solve the equation for y in terms of x, then we write y = f(x) and say that the function is given explicitly. See examples on Pg 127.

20. DETERMINING WHETHER AN EQUATION IS A FUNCTION Determine if x2 + y2 = 1 is a function. This means that for certain values of x, there are two possible outcomes for y. Thus, this is not a function!

21. Important Facts About Functions: 1. For each x in the domain of a function f, there is one and only one image f(x) in the range. For every x, there is only one y. 2. f is the symbol 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. f(x) is another name for y.

22. Important Facts About Functions: 3. 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.

23. DOMAIN OF A FUNCTION If a function is being described symbolically and it comes with a specific domain, that domain should be expressly given. Otherwise, the domain of the function will be assumed to be the “natural domain”.

24. EXAMPLE: f(x) = x 2 Knowing the function of squaring a number, we can determine that the natural domain is all real numbers because any real number can be squared. We can also look at a graph.

25. EXAMPLE: f(x) = x 2 + 5x This is simply a modification of the squaring function. Thus, we can determine that the natural domain is all real numbers. We can also look at a graph.

26. EXAMPLE Find the domain: D: { x½ }

27. EXAMPLE Find the domain: 4 – 3t 0 3t 4

28. EXAMPLE Find the domain : f(x) = x3 + x - 1 All real numbers

29. OPERATIONS ON FUNCTIONS • Notation for four basic operations on functions: • (f + g)(x) = f(x) + g(x) • (f - g)(x) = f(x) - g(x) • (f ·g)(x) = f(x) ·g(x) • (f / g)(x) = f(x) / g(x)

30. OPERATIONS ON FUNCTIONS Do Example 10

31. CONCLUSION OF SECTION 1.1