2. Definition of a Function A function is a set of ordered pairs in which no two ordered pairs of the form (x, y) have the same x-value with different y-values. That sounds easy enough. Maybe we should look at some examples. Example 1 (-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 2) (3, 3)3) This is a function because there are no repeating x-values. Example 2 (5, 7) (3, 5) (1, 4) (0, 0) (1, 6) (4, 2) (6, 3)3) This is not a function because there are repeating x-values. Example 2 is a relation.

3. Representing a Function Suppose a plumber gets paid $40 for traveling time to a job plus $60 for each hour he takes to complete the job. The plumber charges for whole-number hours. The function that shows the relationship between the number of hours, x, that the plumber works and the amount of money, y, that he charges can be represented in different ways. Algebraic Representation Table Representation 1100 2160 3220 Graphical Representation 0 1 2 3 4 5 6 440 400 360 320 280 240 200 160 120 80 40 (1, 100) (2, 160) (3, 220) (4, 280) (5, 340) (6, 400) x hours y dollars 4280 5340 6 400

4. Function Notation For any function f, the notation f(x) is read “ f of x”x” and represents the value of y when x is replaced by the number or expression inside the parenthesis. Input Value Independent Variable f(x)} x}x} Output Value Dependent Variable Domain Set of all independent variables for which a function is defined. All x-values. Range Set of all dependent variables for which a function is defined. All values of f(x). Domain Range

5. Evaluating a Function Evaluate f(3) Evaluate f(-2) Evaluate f(x+3) This is really easy. I better push the easy button.

6. Determining the Domain and Range of a Function What are the domain and range of the following functions? That means find all the x-values for which the function is defined, and all the corresponding y- values. Since the value under the radical sign can’t be negative, the x value can’t be less than 2. With this domain, the value of the function will never be less than 0. Since the denominator can’t be zero, the x value can’t be positive 3 or negative 3. With this domain, the value of the function can be any real number.

7. Vertical Line Test A graph represents a function if any vertical line drawn intersects it in at most one point. The vertical line never intersects the graph in more than one point, therefore this is a function. The vertical line does intersect the graph in more than one point, therefore this is not a function.

8. Graphs of Relations and Functions Is a Function Not a Function Is a Function Not a Function Is a Function Not a Function

9. Equations of Functions Let’s take a look at the equations from the previous page. Equations that are Functions Equations that are not Functions y cannot be squared x cannot be a constant y cannot be absolute value That was easy

10. One-to-One Functions A function is a One-to-One Function if the same value of y is never associated with two different values of x. That means a One-to-One Function cannot have repeating x values or repeating y values. (-3, -9) (-2, -7) (-1, -5) (0, -3) (1, -1) (2, 1) (3, 3)3) This is a One-to-One Function because there are no repeating y values. (-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 4) (3, 9)9) This is not a One-to-One Function because there are repeating y values.

11. Horizontal Line Test A graph represents a one-to-one function if any horizontal line drawn intersects it in at most one point. The horizontal line never intersects the graph in more than one point, therefore this is a one-to-one function. The horizontal line does intersect the graph in more than one point, therefore this is not a one-to-one function.