Sullivan Algebra & Trigonometry: Section 3.1 Functions

1. Sullivan Algebra & Trigonometry: Section 3.1Functions • Objectives • Determine Whether a Relation Represents a Function • Find the Value of a Function • Find the Domain of a Function • Form the Sum, Difference, Product and Quotient of Two Functions

2. Let X and Y be two nonempty sets of real numbers. A function from X into Y is a relation that associates with each element of X a unique element of Y. The set X is called the domain of the function. For each element x in X, the corresponding element y in Y is called the image of x. The set of all images of the elements of the domain is called the range of the function.

3. f x y x y x X Y RANGE DOMAIN

4. Example: Which of the following relations are function? {(1, 1), (2, 4), (3, 9), (-3, 9)} A Function {(1, 1), (1, -1), (2, 4), (4, 9)} Not A Function

5. Functions are often denoted by letters such as f, F, g, G, and others. The symbol f(x), read “f of x” or “f at x”, is the number that results when x is given and the function f is applied. Elements of the domain, x, can be thought of as input and the result obtained when the function is applied can be thought of as output. Restrictions on this input/output machine: 1. It only accepts numbers from the domain of the function. 2. For each input, there is exactly one output (which may be repeated for different inputs).

6. For a function y = f(x), the variable x is called the independent variable, because it can be assigned any of the permissible numbers from the domain. The variable y is called the dependent variable, because its value depends on x. The independent variable is also called the argument of the function.

7. Example: Given the function Find: f (x) is the number that results when the number x is applied to the rule for f. Find:

8. The domain of a function f is the set of real numbers such that the rule of the function makes sense. Domain can also be thought of as the set of all possible input for the function machine. Example: Find the domain of the following function: Domain: All real numbers

9. Example: Find the domain of the following function: Example: Find the domain of the following function:

10. The domain of the function is Example: Express the area of a circle as a function of its radius. The dependent variable is A and the independent variable is r.

11. If f and g are functions, their sum f + g is the function given by (f + g)(x) = f(x) + g(x). The domain of f + g consists of the numbers x that are in the domain of f and in the domain of g. If f and g are functions, their difference f - g is the function given by (f - g)(x) = f(x) - g(x). The domain of f - g consists of the numbers x that are in the domain of f and in the domain of g.

12. Their product is the function given by The domain of consists of the numbers x that are in the domain of f and in the domain of g. Their quotient f / g is the function given by The domain of f / g consists of the numbers x for which g(x) 0 that are in the domain of f and in the domain of g.

13. Example: Define the functions f and g as follows: Find each of the following and determine the domain of the resulting function. a.) (f + g)(x) = f(x) + g(x)

14. b.) (f - g)(x) = f(x) - g(x) c.) ( )(x) = f(x)g(x)

15. d.) We must exclude x = - 4 and x = 4 from the domain since g(x) = 0 when x = 4 or - 4.