1. Digital Lesson An Introduction to Functions

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A relation is a rule of correspondence that relates two sets. For instance, the formula I = 500r describes a relation between the amount of interest I earned in one year and the interest rate r. In mathematics, relations are represented by sets of ordered pairs (x, y). Definition of Relations

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 The set A is called the domain of the function. A function from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set B is called the range of the function. Definition of Function

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Function x y DomainRange Set A Set B Function Diagram

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Example: Determine whether the relation represents y as a function of x. a) {(-2, 3), (0, 0), (2, 3), (4, -1)} b){(-1, 1), (-1, -1), (0, 3), (2, 4)} Function Not a Function Example: Determine Functions

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Functions represented by equations are often named using a letter such as f or g. The symbol f (x), read as “the value of f at x” or simply as “f of x”, is the element in the range of f that corresponds with the domain element x. That is, y = f (x) f(x) : “f of x”

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Input x The domain elements, x, can be thought of as the inputs and the range elements, f (x), can be thought of as the outputs. Output f (x) Function f Inputs & Outputs

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function. Example: Let f (x) = x 2 – 3x – 1. Find f (–2). f (x) = x 2 – 3x – 1 f (–2) = (–2) 2 – 3(–2) – 1 Substitute –2 for x. f (–2) = 4 + 6 – 1 Simplify. f (–2) = 9 The value of f at –2 is 9. Evaluating Functions

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Let f (x) = 4x – x 2. Find f (x + 2). f (x) = 4x – x 2 f (x + 2) = 4(x + 2) – (x + 2) 2 Substitute x + 2 for x. f (x + 2) = 4x + 8 – (x 2 + 4x + 4) Expand (x + 2) 2. f (x + 2) = 4x + 8 – x 2 – 4x – 4 Distribute –1. f (x + 2) = 4 – x 2 The value of f at x + 2 is 4 – x 2. Example: Evaluating Functions

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 The domain of a function f is the set of all real numbers for which the function makes sense. Example: Find the domain of the function f (x) = 3x +5 Domain: All real numbers Definition of Domain

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Find the domain of the function The function is defined only for x-values for which x – 3  0. Solving the inequality yields x – 3  0 x  3 Domain: {x| x  3} Example: Find Domain

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Find the domain of the function The x values for which the function is undefined are excluded from the domain. The function is undefined when x 2 – 1 = 0. x 2 – 1 = 0 (x + 1)(x – 1) = 0 x =  1 Domain: {x| x   1} Example: Find Domain