Section 1.2 Functions Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–1 Section Objectives: Students will know how to evaluate functions and find their domains, and how to evaluate difference quotients

Definition of a Function Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–2 Relation: a set of ordered pairs. Domain: the set of all x-coordinates. Range: the set of all y-coordinates. A function is a relation in which each domain element corresponds to one and only one range element. (X never repeats)

Example 1 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–3 S = {(1, 3), (-5, 4), (7, 3)} a) What is the domain of S? {-5, 1, 7} b) What is the range of S? {3, 4} c) Is S a function? Yes. (1, 3) and (7, 3), when you see two elements from the range (not the domain) this does not violate the definition of a function.

Example 2 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–4 Which of the following equations represents y as a function of x? (hint: solve for y) a. The equation is uniquely solved for y; hence y is a function of x. b. The equation is not uniquely solved for y; hence y is not a function of x.

Function Notation (do not need to write) Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–5 Name the function f. Since we say that y is a function of x, we can replace y with f(x). Now, if we need to know the value of y in the f function when x equals a, all we have to do is write is f(a). This value is read “f of a.” Important Parts to Know: 1) y = f(x). 2) f is the name of the function, not another variable.

Example 3 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–6 Find the following:

Example 4 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–7 a)f(2) = = 3, since 2 > 0 b) f(-5) = -(-5) = 5, since –5 < 0

The Domain of a Function Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–8 Define the implied domain of a function to be the set of all x such that the corresponding y is a real number. At least for a while, we will consider only three situations, 1. Polynomials ⇒ domain is (-∞, ∞). 2. Fractions ⇒ cannot have any number in the domain that makes the denominator zero. 3. Radicals ⇒ if the index is even, then the radicand must be nonnegative.

Example 5 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–9

Difference Quotient Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–10 Evaluate f(x) = 2 + 3x – x 2 a.f(x+h) b.f(x+h) – f(x)

Evaluate a Difference Quotient Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–11

Summary of Function Terminology (Do not need to write) Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–12 Function: A function is a relationship between two variables such that to each value of the independent variable there corresponds exactly one value of the dependent variable Function Notation: y = f(x) f is the name of the function. y is the dependent variable x is the independent variable f(x) is the value of the function at x Domain: all values of x for which the function is defined. Implied Domain: if the domain is not specified, this consists of all real numbers for which the expression is defined. Range: all values of y.

Homework: Pg 25 #30-40E, 56-60E, 82, 84 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–13