Date: 1.2 Functions And Their Properties A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation, and the set of all second components is called the range of the relation. The domain is the set of all first components (the x-values). domain: Find the domain and range of the relation {(1994,56.21), (1995,51.00), (1996,47.70), (1997,42.78), (1998,39.43)} {1994,1995,1996,1997,1998} The range is the set of all second components (the y-values). range: {56.21, 51.00, 47.70, 42.78, 39.43}

Definition of a Function: A function is a relation that assigns to each domain exactly one range. SolutionWe begin by making a figure for each relation that shows the domain and the range. Determine whether each relation is a function. a. {(1, 6), (2, 6), (3, 8), (4, 9)}b. {(6,1),(6,2),(8,3),(9,4)} DomainRange (a) Figure (a) shows that every element in the domain corresponds to exactly one element in the range. No two ordered pairs in the given relation have the same x and different y’s. For every x, there is only one y. Thus, the relation is a function DomainRange (b) Figure (b) shows that 6 corresponds to both 1 and 2 (6 has two y’s). This relation is not a function; two ordered pairs have the same x and different y’s.

Determine whether each equation is a function. a. y = 2x + 3 b. x = y 2 Consider the values of x and y. For every x is there only one y value. x y Function x has two y values Not a Function

x y c. y is a function of x for the graphs in (b) and (c). y is not a function since 2 values of y correspond to an x-value. y is a function of x x y d. y is not a function since 2 values of y correspond to an x-value. The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. Use the vertical line test to identify graphs in which y is a function of x. x y a. x y b.

Finding a Function’s Domain: It’s domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in an even root of a negative number. Find the domain of the function: 1 and -1 need to be excluded from the domain because they cause division by zero.

Find the domain of the function: Numbers that result in an even root of a negative number must be excluded. 2x + 3 must be positive For these values an even root will be positive. Confirm this graphically. Find the range graphically.

Find the domain and range of the function: x = 0 is not in the domain because a denominator cannot be zero. y = 0 is not in the range Find the domain and range graphically. Confirm the domain and range algebraically. y = 0 is not in the range by trying to solve when y = 0

SolutionWe begin by setting up a partial table of coordinates. Graph f (x) = x Use integer values of x from the set {-3, -2, -1, 0, 1, 2, 3}. Use the graph to specify the function's domain and range. Plot the points. (-1, 2)f (-1) = (-1) = 2 (0, 1)f (0) = = 10 (-2, 5)f (-2) = (-2) = 5-2 (-3, 10)f (-3) = (-3) = 10-3 (x, y) or (x, f (x))f (x) = x 2 + 1x (3, 10)f (3) = (3) = 103 (2, 5)f (2) = (2) = 52 (1, 2)f (1) = = 21 Domain: all Reals Range: [1, oo) The points on the graph of f have x-coordinates that extend indefinitely far to the left and to the right. Thus, the domain consists of all real numbers, represented by (-∞, ∞). The points on the graph have y-coordinates that start at 1 and extend indefinitely upward. Thus, the range consists of all real numbers greater than or equal to 1, represented by [1, ∞).

Complete Student Checkpoint Determine whether each relation is a function: a. {(1,2),(3,4),(5,6),(5,8)} b. {(1,2),(3,4),(6,5),(8,5)} Solve each equation for y and then determine whether the equation defines y as a function of x. Not a function Function x y Function x y Not a function

Smooth, Continuous Graphs Two important features of the graphs of polynomial functions are that they are smooth and continuous. By smooth, we mean that the graph contains only rounded curves with no sharp corners. Bycontinuous, we mean that the graph has no breaks and can be drawn without lifting your pencil from the rectangular coordinate system. These ideas are illustrated in the figure. Smooth rounded corner x x y y

This graph has removable discontinuity at x = 1. It is removable because we could define f(1) in such a way as to plug the hole and make f(x) continuous as 1. These graphs have discontinuity. They are not continuous everywhere a.b. Discontinuities This graph has jump discontinuity. This discontinuity is not removable because there is more than just a hole at x = 0; the gap makes it impossible to plug.

This graph has discontinuity. It is not continuous everywhere. Discontinuities This graph has infinite discontinuity. This discontinuity is definitely not removable at x = -2.

Definitions of Local Maximum and Local Minimum A function value f(a) is a local maximum of f if there exists an open interval about a such that f(a) > f(x) for all x in the open interval A function value f(b) is a local minimum of f if there exists an open interval about b such that f(b) < f(x) for all x in the open interval. It occurs at (1, 4), the maximum value is 4 It occurs at (-3, -4), the minimum value is -4 Local is also referred to as Relative or Absolute.

Increasing, Decreasing, and Constant Functions Constant f (x 1 ) = f (x 2 ) (x 1, f (x 1 )) (x 2, f (x 2 )) Increasing f (x 1 ) < f (x 2 ) (x 1, f (x 1 )) (x 2, f (x 2 )) Decreasing f (x 1 ) > f (x 2 ) (x 1, f (x 1 )) (x 2, f (x 2 )) A function is increasing on an interval if for any x 1, and x 2 in the interval, where x 1 < x 2, then f (x 1 ) < f (x 2 ). A function is decreasing on an interval if for any x 1, and x 2 in the interval, where x 1 f (x 2 ). A function is constant on an interval if for any x 1, and x 2 in the interval, where x 1 < x 2, then f (x 1 ) = f (x 2 ).

decreasing on the interval (-∞, 0) increasing on the interval (0, 2) decreasing on the interval (2, ∞) Describe the increasing, decreasing, or constant behavior of each function whose graph is shown a.b. Example The function is defined in two pieces. constant on the interval (-∞, 0) increasing on the interval [0, ∞)

Lower Bound, Upper Bound, and Bounded The function is bounded below if there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f. x y The function is bounded above if there is some number b that is less than or equal to every number in the range of f. Any such number b is called a upper bound of f. bounded below x y not bounded above bounded above not bounded below

Lower Bound, Upper Bound, and Bounded The function is bounded if it is bounded both above and below. The function is: bounded not bounded above not bounded below x y x y

Definition of Even and Odd Functions The function f is an even function if f (-x) = f (x) for all x in the domain of f. The right side of the equation of an even function does not change if x is replaced with -x. The function f is an odd function if f (-x) = -f (x) for all x in the domain of f. Every term in the right side of the equation of an odd function changes sign if x is replaced by -x.

Identify the following function as even, odd, or neither: f(x) = 3x Use the given function equation to find f(-x). f(-x) = 3(-x) 2 -2 = 3x The right side of the equation of the given function did not change when we replaced x with -x. Algebraically: Because f(-x) = f(x) f is an even function.

x y Even Functions and y-Axis Symmetry The graph of an even function in which f (-x) = f (x) is symmetric with respect to the y-axis. Odd Functions and Origin Symmetry The graph of an odd function in which f (-x) = - f (x) is symmetric with respect to the origin. x y

Identify the following function as even, odd, or neither: f(x) = 3x Identify graphically. Because f(x) looks the same on either side of the y-axis f is an even function.

Determine algebraically whether each of the following functions is even, odd, or neither. does not change EVEN all terms change signs ODD all terms do not change signs NEITHER

Arrow Notation Symbol Meaning x  a  x approaches a from the right. x  a  x approaches a from the left. x   x approaches infinity; that is, x increases without bound. x    x approaches negative infinity; that is, x decreases without bound.

The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. f (x)   as x  a  f (x)   as x  a  The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. f (x)   as x  a  f (x)   as x  a  f a y x x = a f a y x Definition of a Vertical Asymptote

The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. x = a f a y x f a y x f (x)  as x  a  f (x)    as x  a  Definition of a Vertical Asymptote

Locating Vertical Asymptotes If f(x) = p(x) / q(x) is a rational function in which p(x) and q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f. 5 is a zero of x - 5, therefore x = 5 is a vertical asymptote of f Find the vertical asymptote of this function:

Find the vertical asymptotes, if any, of the graph of each rational function. a. b. no vertical asymptotes

The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound. f y x y = b x y f f y x f(x)  b as x  f(x)  b as x  f(x)  b as x  Definition of a Horizontal Asymptote

Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. 1.If n<m, the x-axis, or y=0, is the horizontal asymptote of the graph of f. 2.If n=m, the line y = a n /b m is the horizontal asymptote of the graph of f. 3.If n>m, the graph of f has no horizontal asymptote.

Find the horizontal asymptotes, if any, of the graph of each rational function. a. b. c. If n<m, the x-axis, or y = 0 If n>m, no horizontal asymptote If n=m, then y = a/b, or y = 3

Strategy for Graphing a Rational Function Suppose that f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions with no common factors. 1. Determine whether the graph of f has symmetry. f (  x)  f (x): y-axis symmetry f (  x)  f (x): origin symmetry 2.Find the y-intercept (if there is one) by evaluating f (0). 3.Find the x-intercepts (if there are any) by solving the equation p(x)  0. 4.Find any vertical asymptote(s) by solving the equation q (x)  0. 5.Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

the graph has no symmetry The y-intercept is (0,-3/10) The x-intercept is (3/2, 0) The vertical asymptote is x = -2 The horizontal asymptote is y = 2/5 Test points include (-5, 13/15), (-4, 11/10), (-3, 9/5), (-7/2, 16/5), (-1, -1), (2, 1/20) Sketch the graph of

Identify the intervals on which the function is increasing, decreasing, and constant. x y need to find out value it begins or ends but is not included Constant: Increasing: graphing calculator interpretation

Suppose f is an odd function. For each statement below, explain why the statement is possible, necessarily true, or impossible. a. f(3) = 4 and f(-4) = -3 b. f(3) = 4 and f(-3) = -4 c. f(0) = -2 and f(-0) = 2 Possible For odd functions f (-x) = - f (x), so if (-3,-4) and (4,3) are in the function this statement is possible. Impossible The graph of an odd function in which f (-x) = - f (x) is symmetric with respect to the origin. f(0) = 0 for this to be an odd function Necessarily true For odd functions f (-x) = - f (x), so f is an odd function in this statement.

Matching functions using end behavior Match the functions to their graphs on page 97. When x is very large, the denominator x 2 +1 is almost the same as x 2, replace this to make simpler functions and compare end behaviors to match with graphs. a. matches graph (iv) ending behavior, it approaches y = 0 b. matches graph (iii) end behavior, it approaches y = 3 c. matches graph (ii) end behavior, it approaches y = 3x d. matches graph (i) end behavior, it approaches y = 3x 2

Functions and Their Properties