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Chapter 1 Chapter 8 Graphs, Functions, and Systems of Equations and Inequalities Section 8-4 An Introduction to Functions: Linear Functions, Applications, and Models © 2008 Pearson Addison-Wesley. All rights reserved 8-4-2 © 2008 Pearson Addison-Wesley. All rights reserved An Introduction to Functions: Linear Functions, Applications, and Models • • • • • • • Relations and Functions Domain and Range Graphs of Relations Graphs of Functions Function Notation Linear Functions Modeling with Linear Functions Terminology If the value of the variable y depends on the value of the variable x, the y is the dependent variable and x the independent variable. Independent variable Dependent variable (x, y) 8-4-3 8-4-4 © 2008 Pearson Addison-Wesley. All rights reserved © 2008 Pearson Addison-Wesley. All rights reserved Relation Relation Relations can be defined by equations. A relation is a set of ordered pairs. For example, the sets y = x + 1 is the set of all ordered pairs (x, y) where the second coordinate (y) is one greater than the first coordinate (x). F = {(1, 2), (–1, 5), (4, 3)} and G = {(1, 5), (9, 0), (9, 8)} are both relations. 8-4-5 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-6 © 2008 Pearson Addison-Wesley. All rights reserved 1 Function Domain and Range A function is a relation in which for each value of the first component of the ordered pairs there is exactly one value of the second component. In a relation, the set of all values of the independent variable (x) is the domain. The set of all values of the dependent variable (y) is the range. Of the two sets, F = {(1, 2), (–1, 5), (4, 3)} and G = {(1, 5), (9, 0), (9, 8)}, only F is a function. 8-4-7 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-8 © 2008 Pearson Addison-Wesley. All rights reserved Example: Determining Domain and Range Graphs of Relations y The graph of a relation is the graph of its ordered pairs. The graph gives a picture of the relation, which can be used to determine its domain and range. y x x Domain {-3, -2, 0, 1} Domain -1 ≤ x ≤ 1 Range {-2, -1, 2, 3} Range -3 ≤ y ≤ 3 8-4-9 © 2008 Pearson Addison-Wesley. All rights reserved Example: Determining Domain and Range y 8-4-10 © 2008 Pearson Addison-Wesley. All rights reserved y x Interval notation Domain and range can be expressed using interval notation. x Domain is all x Domain is all x Range is all y Range -3 ≤ y ≤ ∞ Recall that for our ellipse: Domain -1 ≤ x ≤ 1 Range -3 ≤ y ≤ 3 Using interval notation: x ∈ [-1, 1], y ∈ [-3, 3] 8-4-11 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-12 © 2008 Pearson Addison-Wesley. All rights reserved 2 Agreement on Domain Interval notation [a, b] means all values from a to b, including the endpoints. Same as a ≤ x ≤ b The domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable. (a, b) means all values from a to b, not including the endpoints. Same as a < x < b 8-4-13 8-4-14 © 2008 Pearson Addison-Wesley. All rights reserved © 2008 Pearson Addison-Wesley. All rights reserved Graphs of Functions Vertical Line Test In a function each value of x leads to only one value of y, so any vertical line drawn through the graph of a function must intersect the graph in at most one point. This is called the vertical line test for a function. If a vertical line intersects the graph of a relation in more than one point, then the relation is not a function. 8-4-15 8-4-16 © 2008 Pearson Addison-Wesley. All rights reserved © 2008 Pearson Addison-Wesley. All rights reserved Example: Determining Whether a Relation is a Function Example: Vertical Line Test y y x Not a function – the same x-value corresponds to multiple y-values Determine whether each equation defines a function. If it is a function, give the domain. x Function – each x-value corresponds to only one y-value a) y = x +1 b) y<x 8-4-17 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-18 © 2008 Pearson Addison-Wesley. All rights reserved 3 Example: Determining Whether a Relation is a Function Variations of the Definition of Function Solution 1. a) A function; since any real x input returns a real y output, the domain is all real x. A function is a relation in which for each value of the first component of the ordered pairs there is exactly one value of the second component. 2. A function is a set of distinct ordered pairs in which no first component is repeated. 3. A function is a rule or correspondence that assigns exactly one range value to each domain value. b) Not a function. 8-4-19 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-20 © 2008 Pearson Addison-Wesley. All rights reserved Function Notation Function Notation When a function f is defined with a rule or an equation using x and y for the independent and dependent variables, we say “y is a function of x” to emphasize that y depends on x. We use the notation Note that f (x) is just another name for the dependent variable y. If f (x) = 3x + 1, we find f (2) by replacing x with 2, f (2) = 3(2) + 1 = 7 y = f (x), called function notation, to express this and read f (x) as “f of x.” For example if y = 3x + 1, we write f (x) = 3x + 1. Read f (2) as “f of 2” or “f at 2.” 8-4-21 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-22 © 2008 Pearson Addison-Wesley. All rights reserved Example: Using Function Notation Linear Functions Let f (x) = x 2 + 3x – 1. Find the following. a) f (2) b) f (-2) c) f (0) d) f (2x) Solution A function that can be written in the form f (x) = mx + b for real numbers m and b is a linear function. a) f (2) = 2 2 + 3(2) – 1 = 9 b) f (-2) = (-2)2 +3(-2) – 1 = -3 c) f (0) = 0 + 0 – 1 = –1 d) f (2x) = (2x) 2 + 3(2x) – 1 = 4x 2 + 6x – 1 8-4-23 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-24 © 2008 Pearson Addison-Wesley. All rights reserved 4 Example: Graphing Linear Functions Graphing a Linear Function An easy way to graph a linear function is to find 2 ordered pairs that satisfy the function. Graph each linear function. a) f (x) = –2x + 1 b) f (x) = 2 Solution Plot those ordered pairs, then connect with a line. y y x 8-4-25 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-26 © 2008 Pearson Addison-Wesley. All rights reserved Linear Cost Models Linear Cost Models For linear cost function Linear functions can be good models for a company’s costs. Fixed cost or up-front cost, is the cost required before production can begin. Variable cost is the cost to produce each item. C(x) = mx + b m is the variable cost, b is the fixed cost, x is the number of items produced. 8-4-27 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-28 © 2008 Pearson Addison-Wesley. All rights reserved Example: Modeling with Linear Functions Linear Revenue Model A company produces DVDs of live concerts. The company pays $200 for advertising the DVDs. Each DVD costs $12 to produce and the company charges $20 per disk. A company’s revenues can be modeled R(x) = px where p is the price per item, x is the number of items sold. a) Express the cost C as a function of x, the number of DVDs produced. b) Express the revenue R as a function of x, the number of DVDs sold. c) When will the company break-even? That is, for what value of x does revenue equal cost? 8-4-29 © 2008 Pearson Addison-Wesley. All rights reserved x 8-4-30 © 2008 Pearson Addison-Wesley. All rights reserved 5 Example: Modeling with Linear Functions Example: Modeling with Linear Functions Solution Solution (continued) a) The fixed cost is $200 and for each DVD produced, the variable cost is $12. The cost C can be expressed as a function of x, the number of DVDs produced: C(x) = 12x + 200. c) The company will just break even (no profit and no loss) as long as revenue just equals cost, or C(x) = R(x). This is true whenever 12x + 200 = 20x 200 = 8x 25 = x. If 25 DVDs are produced and sold, the company will break even. b) Each DVD sells for $20, so revenue R is given by: R(x) = 20x. 8-4-31 © 2008 Pearson Addison-Wesley. All rights reserved 8-4-32 © 2008 Pearson Addison-Wesley. All rights reserved 6