1. Graphs of Functions Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of a function f is the collection of ordered pairs (x, f(x)) where x is in the domain of f. Definition of Graph x y 4 -4 (2, –2) is on the graph of f(x) = (x – 1) 2 – 3. (2, –2) f(2) = (2 – 1) 2 – 3 = 1 2 – 3 = – 2

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 x y 4 -4 The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists. The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain. Domain Range Domain & Range

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 x y – 1 1 Example: Find the domain and range of the function f (x) = from its graph. The domain is [–3,∞). The range is [0,∞). Range Domain Example: Domain & Range (–3, 0)

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 x y 4 -4 Vertical Line Test A relation is a function if no vertical line intersects its graph in more than one point. Vertical Line Test This graph does not pass the vertical line test. It is not a function. This graph passes the vertical line test. It is a function. y = x – 1 x = | y – 2| x y 4 -4

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 decreasing on an interval if, for any x 1 and x 2 in the interval, x 1 f (x 2 ), constant on an interval if, for any x 1 and x 2 in the interval, f (x 1 ) = f (x 2 ). The graph of y = f (x): increases on ( – ∞, – 3), decreases on ( – 3, 3), increases on (3, ∞). Increasing, Decreasing, and Constant Functions A function f is: increasing on an interval if, for any x 1 and x 2 in the interval, x 1 < x 2 implies f (x 1 ) < f (x 2 ), (3, – 4) x y ( – 3, 6) –2–2 2

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Minimum and Maximum Values A function value f(a) is called a relative minimum of f if there is an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) f(x). x y A function value f(a) is called a relative maximum of f if there is an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) f(x). Relative minimum Relative maximum

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Graphing Utility: Approximating a Relative Minimum Graphing Utility: Approximate the relative minimum of the function f(x) = 3x 2 – 2x – 1. – 6 6 6 – 0.86 – 4.79 – 1.79 2.14 0.580.76 -3.24 -3.43 Zoom In: The approximate minimum is (0.67, –3.33).

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 x y 4 -4 A piecewise-defined function is composed of two or more functions. Piecewise-Defined Functions f(x) = 3 + x, x < 0 x 2 + 1, x 0 Use when the value of x is less than 0. Use when the value of x is greater or equal to 0. (0 is not included.) open circle (0 is included.) closed circle

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 A function f is even if for each x in the domain of f, f (– x) = f (x). Even Functions x y f (x) = x 2 f (– x) = (– x) 2 = x 2 = f (x) f (x) = x 2 is an even function. Symmetric with respect to the y-axis.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 A function f is odd if for each x in the domain of f, f (– x) = – f (x). Odd Functions x y f (x) = x 3 f (– x) = (– x) 3 = –x 3 = – f (x) f (x) = x 3 is an odd function. Symmetric with respect to the origin.