Functions. MATH 160, Precalculus. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Functions

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1 Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011

2 Objectives In this lesson we will learn to: determine whether relations between variables are functions, use function notation and to evaluate functions, find the domains of functions, use functions in applications to real-life problems, evaluate difference quotients.

and to evaluate functions, find the domains of functions, use

3 Functions We frequently encounter quantities that are related to each other through some relation or rule of correspondence. Definition A function f 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 A is called the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs). Variable x is called the independent variable and variable y is called the dependent variable.

Definition A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one

4 Functions We frequently encounter quantities that are related to each other through some relation or rule of correspondence. Definition A function f 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 A is called the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs). Variable x is called the independent variable and variable y is called the dependent variable. Notation: f : A B

5 Characteristics of Functions 1 Each element of A must be matched with an element of B. 2 Some elements of B may not be matched with any element of A. 3 Two or more elements in A may be matched with the same element in B. 4 An element in A (the domain) cannot be matched with two different elements in B.

3 Two or more elements in A may be matched with the same element in B.

6 Representation of Functions 1 Verbally by a sentence that describes how the input variable is related to the output variable. 2 Numerically by a table or list of ordered pairs that matches input values with output values. 3 Graphically by points on a graph in a coordinate plane in which the input values are represented by the horizontal axis and the output values are represented by the vertical axis. 4 Algebraically by an equation in two variables.

3 Graphically by points on a graph in a coordinate plane in which the input values are represented by the

7 Examples Determine whether the following are representations of functions. Input x Table, Output y Ordered pairs, {(1, a), (0, a), (2, c), (3, b)} 3 Sentence, The input variable is the year and the output variable is the number of hurricanes observed between June 1 and October 1. 4 Equation, x 2 + y 2 = 1.

(2, c), (3, b)} 3 Sentence, The input variable is the year and the output variable

8 Function Notation and Evaluation Often functions are given letter names (for example f ). In this case the equation describing the function will use the symbols f (x) (read as f of x) in place of y. f (x) = x To evaluate the function f (x) we replace x in the equation by some value from the domain of the function.

In this case the equation describing the function will use the symbols f (x)

9 Examples Evaluate the following functions. f (1) f (6) g( 1) g(4) g(0) f (x) = x 2 3x + 2 x 3 1 if x < 0 g(x) = 5 if x = 0 3 x 2 if x > 0

10 Examples Let f (x) = x 2 3x + 2 g(x) = x Find all the value of x for which f (x) = 0. Find all the values of x for which f (x) = g(x).

11 Examples Let f (x) = x 2 3x + 2 g(x) = x Find all the value of x for which f (x) = 0. x 2 3x + 2 = 0 (x 1)(x 2) = 0 x = 1 and x = 2 Find all the values of x for which f (x) = g(x).

12 Examples Let f (x) = x 2 3x + 2 g(x) = x Find all the value of x for which f (x) = 0. x 2 3x + 2 = 0 (x 1)(x 2) = 0 x = 1 and x = 2 Find all the values of x for which f (x) = g(x). x 2 3x + 2 = x x + 2 = 1 3x = 1 1

13 Domain of a Function Often the domain set of an function will not be explicitly stated. The implied domain of such a function will be the largest set of real numbers for which the function expression is defined.

14 Domain of a Function Often the domain set of an function will not be explicitly stated. The implied domain of such a function will be the largest set of real numbers for which the function expression is defined. Find the implied domains of the following functions. f : {(0, 3), (4, 1), (2, 0), ( 1, 4)} { 1, 0, 2, 4} f (x) = 1 x 2 1 g(x) = 3 + 4x Surface area of a sphere S = 4πr 2

function expression is defined. Find the implied domains of the following functions.

15 Domain of a Function Often the domain set of an function will not be explicitly stated. The implied domain of such a function will be the largest set of real numbers for which the function expression is defined. Find the implied domains of the following functions. f : {(0, 3), (4, 1), (2, 0), ( 1, 4)} { 1, 0, 2, 4} f (x) = 1 x 2 {x : x ±1} 1 g(x) = 3 + 4x {x : x 3/4} Surface area of a sphere S = 4πr 2 {r : r > 0}

expression is defined. Find the implied domains of the following functions.

16 Application A ball is thrown by a person and the height of the ball (in feet) t seconds after the ball is thrown is given by y = 16t t Graph the height of the ball and estimate its maximum height. 2 Determine the time at which the ball strikes the ground.

16t 2 + 40t + 5 1 Graph the height of the ball and estimate its

17 Application A ball is thrown by a person and the height of the ball (in feet) t seconds after the ball is thrown is given by y = 16t t Graph the height of the ball and estimate its maximum height. Maximum height: 30 feet 2 Determine the time at which the ball strikes the ground.

1 Graph the height of the ball and estimate its maximum height.

18 Application A ball is thrown by a person and the height of the ball (in feet) t seconds after the ball is thrown is given by y = 16t t Graph the height of the ball and estimate its maximum height. Maximum height: 30 feet 2 Determine the time at which the ball strikes the ground. 16t t + 5 = 0 t = 40 ± (40) 2 4( 16)(5) s

19 Application The inventor of a new game believes that the variable cost for producing the game is $0.95 per unit and the fixed costs are $6,000. The inventor sells each game for $1.69. Let x be the number of games sold. 1 The total cost for the business is the sum of the fixed and variable costs. Write the total cost C as a function of the number of games sold. 2 Write the average cost per unit C = C/x as a function of x.

20 Application The inventor of a new game believes that the variable cost for producing the game is $0.95 per unit and the fixed costs are $6,000. The inventor sells each game for $1.69. Let x be the number of games sold. 1 The total cost for the business is the sum of the fixed and variable costs. Write the total cost C as a function of the number of games sold. C = x 2 Write the average cost per unit C = C/x as a function of x.

21 Application The inventor of a new game believes that the variable cost for producing the game is $0.95 per unit and the fixed costs are $6,000. The inventor sells each game for $1.69. Let x be the number of games sold. 1 The total cost for the business is the sum of the fixed and variable costs. Write the total cost C as a function of the number of games sold. C = x 2 Write the average cost per unit C = C/x as a function of x. C = C x = x x = 6000 x

22 Difference Quotients In calculus you will often evaluate a difference quotient. If h 0, f (x + h) f (x) h

23 Difference Quotients In calculus you will often evaluate a difference quotient. If h 0, f (x + h) f (x) h Example Evaluate the difference quotient for f (x) = x 2 3x + 2.

24 Difference Quotients In calculus you will often evaluate a difference quotient. If h 0, f (x + h) f (x) h Example Evaluate the difference quotient for f (x) = x 2 3x + 2. f (x + h) f (x) h = (x + h)2 3(x + h) + 2 (x 2 3x + 2) h = x 2 + 2xh + h 2 3x 3h + 2 x 2 + 3x 2 h = 2xh + h2 3h h = 2x + h 3

25 Homework Read Section 1.4. Exercises: 1, 5, 9, 13,..., 109, 113

Week 1: Functions and Equations

Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter