1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 10 Graphing Equations and Inequalities

2. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.6 Introduction to Functions

3. Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Vocabulary An equation in 2 variables defines a relation between the two variables. A set of ordered pairs is also called a relation. The domain is the set of x-coordinates of the ordered pairs. The range is the set of y-coordinates of the ordered pairs.

4. Martin-Gay, Developmental Mathematics, 2e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)}. Domain is the set of all x-values; {4, –4, 2, 10}. Range is the set of all y-values; {9, 3, –5}. Example

5. Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Some relations are also functions. A function is a set of order pairs in which each x-coordinate has exactly one y-coordinate. Functions

6. Martin-Gay, Developmental Mathematics, 2e 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Is the relation{(4,9), (–4,9), (2,3), (10, –5)}, also a function? Since each element of the domain is paired with only one element of the range, it is a function. Note: It’s okay for a y-value to be assigned to more than one x-value, but an x-value cannot be assigned to more than one y-value (has to be assigned to ONLY one y-value). Example

7. Martin-Gay, Developmental Mathematics, 2e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Graphs can be used to determine if a relation is a function. Vertical Line Test If a vertical line can be drawn so that it intersects a graph more than once, the graph is not the graph of a function. (If no such vertical line can be drawn, the graph is that of a function.) Vertical Line Test

8. Martin-Gay, Developmental Mathematics, 2e 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Use the vertical line test to determine whether the graph to the right is the graph of a function. x y Since no vertical line will intersect this graph more than once, it is the graph of a function. Example

9. Martin-Gay, Developmental Mathematics, 2e 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Use the vertical line test to determine whether the graph to the right is the graph of a function. x y Since no vertical line will intersect this graph more than once, it is the graph of a function. Example

10. Martin-Gay, Developmental Mathematics, 2e 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Use the vertical line test to determine whether the graph to the right is the graph of a function. Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function. x y Example

11. Martin-Gay, Developmental Mathematics, 2e 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Since the graph of a linear equation is a line, all linear equations are functions, except those whose graph is a vertical line. Thus, all linear equations are functions except those of the form x = c, which are vertical lines. Vertical Line Test

12. Martin-Gay, Developmental Mathematics, 2e 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The variable y is a function of the variable x. For each value of x, there is only one value of y. Thus, we say the variable x is the independent variable because any value in the domain can be assigned to x. The variable y is the dependent variable because its value depends on x. We often use letters such as f, g, and h to name functions. For example, the symbol f(x) means function of x and is read “f of x.” This notation is called function notation. We can use function notation to write the equation y = –3x + 2 as f(x) = –3x + 2. Using Function Notation

13. Martin-Gay, Developmental Mathematics, 2e 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint Note that f(x) is a special symbol in mathematics used to denote a function. The symbol f(x) is read “f of x.” It does not mean f x (f times x).

14. Martin-Gay, Developmental Mathematics, 2e 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. When we want to evaluate a function at a particular value of x, we substitute the x-value into the notation. For example, f(2) means to evaluate the function f when x = 2. So we replace x with 2 in the equation. For our previous example when f(x) = –3x + 2, f(2) = –3(2) + 2 = –6 + 2 = –4. When x = 2, then f(x) = –4, giving us the order pair (2, –4). Function Notation

15. Martin-Gay, Developmental Mathematics, 2e 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Given g(x) = x 2 – 2x, find g(–3). Then write down the corresponding ordered pair. g(–3) = (–3) 2 – 2(–3) = 9 – (–6) = 15. Example The ordered pair is (–3, 15).

16. Martin-Gay, Developmental Mathematics, 2e 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find the domain and the range of the function. x y Domain: – 3 ≤ x ≤ 4] Domain Range: – 4 ≤ x ≤ 2] Range Example

17. Martin-Gay, Developmental Mathematics, 2e 17 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find the domain and the range of the function graphed. x y Domain: all real numbers Domain Range: y ≥ –2 Range Example