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Sections 3.1 and 3.2 Relations and Functions A Refresher on Set Theory Experience says that a brief refresher on some basic notions is welcome, if not completely necessary, at this stage. To that end, we present a brief summary of ‘set theory’ and some of the associated vocabulary and notations we use in the text. Like all good Math books, we begin with a definition. A Refresher on Set Theory Definition: A set is a well-defined collection of objects which are called the ‘elements’ of the set. Here, ‘well-defined’ means that it is possible to determine if something belongs to the collection or not, without prejudice. A Refresher on Set Theory For example, the collection of letters that make up the word “smolko” is well-defined and is a set, but the collection of the worst math teachers in the world is not well-defined, and so is not a set. In general, there are three ways to describe sets. They are, Ways to Describe Sets 1. The Verbal Method: Use a sentence to define a set. 2. The Roster Method: Begin with a left brace { , list each element of the set only once and then end with a right brace }. 3. The Set-Builder Method: A combination of the verbal and roster methods using a ‘dummy variable’ such as x. Ways to Describe Sets For example, let S be the set described verbally as the set of letters that make up the word ‘smolko’. A roster description of S would be {s, m, o, l, k}. Note that we listed ‘o’ only once, even though it appears twice in ‘smolko’. Also, the order of the elements does not matter, so {k, l, m, o, s} is also a roster description of S. Ways to Describe Sets A set-builder description of S is: {x | x is a letter in the word ‘smolko’} The way to read this is: The set of all elements x such that x is a letter in the word ‘smolko’. In each of the above cases, we may use the familiar equals sign and write S {s, m, o, l, k} or S {x | x is a letter in the word ‘smolko’} Ways to Describe Sets S {x | x is a letter in the word ‘smolko’ } {s, m, o, l, k} Clearly m is in S and q is not in S. We express these sentiments mathematically by writing m S and q S. Sets of Numbers 1. The Empty Set: {}{x| x x}. This is the set with no elements. Like the number ‘0’, it plays a vital role in mathematics. 2. The Natural Numbers: {1, 2, 3, …}. 3. The Integers: {…, 3, 2, 1, 0, 1, 2, 3, …}. Sets of Numbers 4. The Rational Numbers: {a/b | a and b and b 0}. It turns out that another way to describe the rational numbers is: {x | x possesses a repeating or terminating decimal representation}. Sets of Numbers 5. The Real Numbers: {x | x possesses a decimal representation}. 6. The Irrational Numbers: {x | x is a non-rational real number}. Said another way, an irrational number is a number whose decimal representation neither repeats nor terminates. Sets of Numbers 7. The Complex Numbers: {x | x a+bi and a,b and i 1 }. Despite their importance, the complex numbers play only a minor role in the text. Interval Notation For the most part, this course focuses on sets whose elements come from the real numbers . Recall that we may visualize as a line. Segments of this line are called intervals of numbers. Below is a summary of the so-called interval notation associated with given sets of numbers. Interval Notation Examples As an example, consider the sets of real numbers described below. Intersection and Union of Sets We will often have occasion to combine sets. There are two basic ways to combine sets: intersection and union. We define both of these concepts below. Definition: Suppose A and B are sets. The intersection of A and B is defined to be the set A B {x | x A and x B }. The union of A and B is defined to be the set A B {x | x A or x B or both}. Intersection and Union of Sets Examples: If A [5,3) and B (1,), then we can find A B and A B graphically. To find A B , we shade the overlap of the two and obtain A B (1,3). To find A B, we shade each of A and B and describe the resulting shaded region to find A B [5,). Intersection and Union of Sets If A [5,3) and B (1,), then we can find A B and A B graphically. More Examples Express the following sets using interval notation. 1. {x | x 2 or x 2} 2. {x | x 3} 3. {x | x 3} 4. {x | 1< x 3 or x 5} Solutions 1. {x | x 2 or x 2} Solutions 2. {x | x 3} Solutions 3. {x | x 3} Solutions 4. {x | 1< x 3 or x 5} The Cartesian Coordinate Plane In order to visualize the pure excitement that is Pre-calculus, we need to unite Algebra and Geometry. Simply put, we must find a way to draw algebraic things. Let us start with possibly the greatest mathematical achievement of all time: The Cartesian Coordinate Plane. The Cartesian Coordinate Plane Imagine two real number lines crossing at a right angle at 0 as drawn below. The Cartesian Coordinate Plane The horizontal number line is usually called the xaxis while the vertical number line is usually called the y-axis. As with the usual number line, we imagine these axes extending o indefinitely in both directions. The Cartesian Coordinate Plane Having two number lines allows us to locate the positions of points off of the number lines as well as points on the lines themselves. For example, consider the point P on the next slide. To use the numbers on the axes to label this point, we imagine dropping a vertical line from the x-axis to P and extending a horizontal line from the y-axis to P. The Cartesian Coordinate Plane This process is sometimes called ‘projecting’ the point P onto the x- (respectively y-) axis. The Cartesian Coordinate Plane The projections of P onto the x and y axis are called respectively, the x and the y coordinates of P. The Cartesian Coordinate Plane This way P is represented by the ordered pair of numbers x 2 and y 4. We write P (2,4). The Cartesian Coordinate Plane In general, any point in the plane is an ordered pair on real numbers. P (x,y). P (x , y) The Cartesian Coordinate Plane Therefore, the plane itself is the set of all possible pairs P (x,y) of real numbers. P (x , y) The Cartesian Coordinate Plane The axes divide the plane into four regions called quadrants. They are labeled with Roman numerals and proceed counterclockwise around the plane: Relations and Functions Relations In certain sense, all of Pre-calculus can be thought of as studying sets of points in the plane. With the Cartesian Plane now fresh in our memory we can discuss those sets in more detail and as usual, we begin with a definition. Definition: A relation is a set of points, (ordered pairs) in the plane. Relations Since relations are sets, we can describe them using the techniques presented before. That is, we can describe a relation verbally, using the roster method, or using set-builder notation. Since the elements in a relation are points in the plane, we often try to describe the relation graphically or algebraically as well. Relations Depending on the situation, one method may be easier or more convenient to use than another. As an example, consider the relation R {(2,1), (4,3), (0,3)} As written, R is described using the roster method. Since R consists of points in the plane, we follow our instinct and plot the points. Relations Doing so produces the graph of R. Examples of Relations Graph the following relations. Solutions The graph for examples 1 and 2 are, Solutions The graph for example 3 is, Solutions The graph for example 4 is, Solutions The graph for example 5 is, Solutions The graph for example 5 is, Relations defined by Equations In this section, we delve more deeply into the connection between Algebra and Geometry by focusing on graphing relations described by equations. The main idea of this section is the following. Relations defined by Equations The graph of an equation is the set of points which satisfy the equation. That is, a point (x,y) is on the graph of an equation if and only if x and y satisfy the equation. Notice that the graph of an equation is a set of points in the plane and therefore by definition, a relation Relations defined by Equations Here, “x and y satisfy the equation” means “x and y make the equation true”. It is at this point that we gain some insight into the word “relation”. If the equation to be graphed contains both x and y, then the equation itself is what is relating the two variables. Examples More specifically, in the next examples, we will consider the graph of the equation x2 + y3 1. Even though it is not specifically spelled out, what we are doing is graphing the relation R {(x,y) | x2 +y3 1} Examples The points (x,y) we graph belong to the relation R and are necessarily related by the equation x2 + y3 1, since it is those pairs of x and y which make the equation true. Determine whether or not (2,1) is on the graph of x2 + y3 1. Examples To determine whether or not (2,1) is on the graph of x2 + y3 1, we substitute x 2 and y 1 into the equation to see if the equation is satisfied. ? (2) ( 1) 1 2 3 3 1 Hence, (2,1) is not on the graph of x2 + y3 1. Examples We now graph the equation x2 + y3 1. To efficiently generate points on the graph of this equation, we first solve for y in terms of x x y 1 2 3 y 1 x 3 3 2 3 y 1 x 2 y 1 x 2 3 3 Examples We now substitute a value in for x, determine the corresponding value y, and plot the resulting point (x,y). For example, substituting x 3 into the equation yields y 1 x 1 (3) 8 2 3 2 3 2 3 so the point (3 , 2) is on the graph. Examples Continuing in this manner, we generate a table of points which are on the graph of the equation. The points are then plotted in the plane as shown below. Examples Remember, these points constitute only a small sampling of the points on the graph of this equation. To get a better idea of the shape of the graph, we could plot more points until we feel comfortable “connecting the dots”. Doing so would result in a curve similar to the one pictured below on the far left. Examples The graph x2 + y3 1 is given by Intercepts of the graph Definition: Suppose the graph of an equation is given. A point on a graph which is also on the x-axis is called an x-intercept of the graph. A point on a graph which is also on the y-axis is called an y-intercept of the graph. Intercepts of the graph Finding the Intercepts of the Graph of an Equation Given an equation involving x and y, we find the intercepts of the graph as follows: x-intercepts have the form (x , 0); set y 0 in the equation and solve for x. y-intercepts have the form (0 , y); set x 0 in the equation and solve for y. Domain and Range of a Relation Suppose R is a relation. The sets of all x- and ycoordinates of the points in R are given special names which we define below. The set of the x-coordinates of all the points in R is called the domain of R. The set of the y-coordinates of all the points in R is called the range of R. Examples of Domain and Range Find the domain and range of the following relations and express them in interval notation whenever possible Functions Introduction to Functions One of the core concepts in College Algebra is that of function. There are many ways to describe a function and we begin by defining a function as a special kind of relation. Definition: A relation F in which each x-coordinate is matched with only one y-coordinate is said to describe y as a function of x. Introduction to Functions In other words, a function F is a relation with the particular property that no two pairs of points in the F have the same x-coordinate. That is, in order to say y is a function of x, we just need to ensure the same x-coordinate is not used in more than one point. Introduction to Functions Which of the following relations describe y as a function of x? A quick scan of the points in R1 reveals that the x-coordinate 1 is matched with two different y-coordinates: namely 3 and 4. Hence in R1, y is not a function of x, or, R1 is not a function. Introduction to Functions Which of the following relations describe y as a function of x? On the other hand, every x-coordinate in R2 occurs only once which means each x-coordinate has only one corresponding y-coordinate. So, R2 does represent y as a function of x, or, R2 is a function. Introduction to Functions In order to see what the concept of function means geometrically, we graph R1 and R2 in the plane. Introduction to Functions The fact that the x-coordinate 1 is matched with two different y-coordinates in R1 presents itself graphically as the points (1,3) and (1,4) lying on the same vertical line, x 1. Introduction to Functions If we turn our attention to the graph of R2, we see that no two points of the relation lie on the same vertical line. Vertical Line Test The Vertical Line Test: A set of points in the plane represents y as a function of x if and only if no two points lie on the same vertical line. In other words, a relation R represents y as a function of x if and only if no two points in R lie on the same vertical line. Vertical Line Test Use the Vertical Line Test to determine which of the following relations describes y as a function of x. Vertical Line Test Use the Vertical Line Test to determine which of the following relations describes y as a function of x. Domain and Range of a Function Suppose F is a relation that defines y as a function of x. That is, F is a function. Then, as before we have, The set of the x-coordinates of all the points in F is called the domain of F. The set of the y-coordinates of all the points in F is called the range of F. Domain and Range of a Function Example: Find the domain and range of the function F {(3, 2), (0,1), (4,2), (5,2)} Solution: The domain of F is the set of the x- coordinates of the points in F, namely, Dom F {3, 0, 4, 5} and the range of F is the set of the y-coordinates, Namely, Ran F {1, 2} Domain and Range of a Function Example: Find the domain and range of the Function G whose graph is given below. Domain and Range of a Function To determine the domain and range of G, we need to determine which x and y values occur as coordinates of points on the given graph. To find the domain, it may be helpful to imagine collapsing the curve to the x-axis and determining the portion of the x-axis that gets covered. This is called projecting the curve onto the x-axis. Domain and Range of a Function Before we start projecting, we need to pay attention to two subtle notations on the graph: the arrowhead on the lower left corner of the graph indicates that the graph continues to curve downwards to the left forever more; and the open circle at (1,3) indicates that the point (1,3) is not on the graph, but all points on the curve leading up to that point are. Domain and Range of a Function Dom G {x | x <1} (,1) Domain and Range of a Function Ran G {x | x 4} (,4] Remark about Functions All functions are relations, but not all relations are functions. Thus the equations which described the relations in previous examples may or may not describe y as a function of x. The algebraic representation of functions is the most important way to view them so we need a process for determining whether or not an equation of a relation represents a function. Three Examples Determine which equations represent y as a function of x. 1. x3 + y2 1 2. x2 + y3 1 3. x2y 1 3y Solution to Example 1 1. x3 + y2 1 Notice that in this case we are referring to the relation defined by R {(x,y) | x3 +y2 1} We solve for y and determine whether each choice of x will determine only one corresponding value of y. Solution to Example 1 Solving for y in terms of x, we get x y 1 3 2 y 1 x 2 3 2 y 1 x 3 y 1 x 3 y 1 x 3 Solution to Example 1 If we substitute x 0 into our equation for y, we get y 1, so that (0,1) and (0,1) are on the graph of this equation. Hence, this equation does not represent y as a function of x. Or equivalently, the relation R {(x,y) | x3 +y2 1} is not a function. Solution to Example 1 The graph of the relation R {(x,y) | x3 +y2 1} is given by Solution to Example 2 2. x2 + y3 1 Notice that in this case we are referring to the relation defined by f {(x,y) | x2 + y3 1} We solve for y and determine whether each choice of x will determine only one corresponding value of y. Solution to Example 2 Solving for y in terms of x, we get x y 1 2 3 y 1 x 3 2 y 1 x 3 2 For every choice of x, the equation returns only one value of y. Hence, this equation describes y as a function of x. Solution to Example 2 The graph of the function f {(x,y) | x2 + y3 1} is given by Solution to Example 3 3. x2y 1 3y Notice that in this case we are referring to the relation defined by g {(x,y) | x2y 1 3y} We solve for y and determine whether each choice of x will determine only one corresponding value of y. Solution to Example 3 x y 1 3y 2 Solving for y we get x 2 3 y 1 1 y 2 x 3 For every choice of x, the equation returns only one value of y. Hence, this equation describes y as a function of x. Solution to Example 3 The graph of the function g {(x,y) | x2y 1 3y} is given by Function Notation A function is a special kind of relation, one in which each x-coordinate is matched with only one y-coordinate. Here is another way of saying it: A real-valued function f is a relation that assigns to each real number x in a set X of numbers, a unique real number y in a second set Y of numbers. The set X is called the domain of the function f and the second set Y is called the codomain of f. Function Notation For each element x in the domain X of the function, the corresponding element y in Y is called the image of x under the function f. The image is denoted by f (x), that is, y = f (x). f (x) is read “f of x.” The set of all images of the elements of the domain is called the range of the function. A way to picture a function is by an arrow diagram f x y x y x X DOMAIN Y Not in the range of f RANGE Function Notation Any symbol that represents an arbitrary number in the domain of a function f is called an independent variable. Any symbol that represents a number in the range of f is called a dependent variable. A function, or relation in general, can be specified: algebraically: by means of a formula numerically: by means of a table graphically: by means of a graph Note on Domains The domain of a function is not always specified explicitly. If no domain is specified for the function f, we take the domain to be the largest set of numbers x for which f (x) makes sense. This "largest possible domain" is sometimes called the natural domain or implied domain. Algebraically Defined Function Is a function represented by a formula. It has the format y f (x) “expression in x” 2 f ( x ) 3 x 2 is a function. Example: f (5) 3(5) 2 77 Substitute 5 for x f ( x h) 3 x h 2 Substitute x+h for x 2 2 3x 6xh 3h 2 2 2 Algebraically Defined Function Is a function represented by a formula. It has the format y f (x) “expression in x” 2 f ( x ) 3 x 2 is a function. Example: In this case the natural domain of the function is the set of all real numbers. That is, Dom f (– , ) Algebraically Defined Function 4 is a function. s(t ) Example: t 1 In this case the natural domain of the function is the set Dom s t | t 1 0 t | t 1 In interval notation this is Dom s ( ,1) (1, ) Algebraically Defined Function Example: h( z ) 2 3z is a function. In this case the natural domain of the function consists of all values of z such that 2 3z 0 or 3z 2 or z 2/ 3 In interval notation this is Dom h [ 2 / 3, ) Numerically Specified Function This is the case when we give numerical values for the function (the outputs, say the y-values) for certain values of the independent variable, say x. In this case the function is represented by a table which looks like. x-values x1 y = f (x) f (x1) x2 … … xn f (x2) … … f (xn) Numerically Specified Function Notice that the table x-values x1 y = f (x) f (x1) x2 … … xn f (x2) … … f (xn) defining the function f is the relation f {(x1 , f (x1)), (x2 , f (x2)), … (xn , f (xn))} written in row format, where the first row is the domain of f and the second row is the range of f. Numerically Specified Function Example: Suppose that the function f is specified by the following table. x f (x) 0 1 2 3.01 -1.03 2.22 3.7 4 0.01 1 Then, f (0) is the value of the function when x = 0. Thus f (0) = 3.01 Look on the table where x = 0 f (1) = 1.03 Look on the table where x = 1 and so on Numerically Specified Function Example: The human population of the world P depends on the time t. The table gives estimates of the world population P (t) at time t, for certain years. For instance, P(1950) 2,560, 000, 000 However, for each value of the time t, there is a corresponding value of P, and we say that P is a function of t. Numerically Specified Function Example: The human population of the world P depends on the time t. Numerically Specified Function Example: The data represents the velocity V of an object, in feet/sec, after t seconds have elapsed. t 0 1 2 3 4 V(t) 2.2 3.55 4.9 6.25 7.6 Note: at 2 seconds the object is going at 4.9 ft/sec, that is V(2) = 4.9 ft/sec. The table can be represented graphically as follows Numerically Specified Function V(t) ft/sec 8 7 6 5 4 3 2 1 -1 1 -1 2 3 4 5 6 7 t in seconds 8 Implicit Form of a Function Implicit Form Explicit Form F(x,y) constant y f(x) Graphs of Functions Obtaining Information from or About the Graph of a Function Graphically Specified Function In applications, the graph of a function often demonstrates more clearly the relationship between the independent variable x and the dependent variable y. Recall that, The graph of a function is the set of all points (x, f (x)) in the xy-plane such that x is in the domain of f . Sometimes the function is only known through its graph and may be very difficult to represent it algebraically. The next example illustrates this case. Graphically Specified Function The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. The figure shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a. Graphically Specified Function Example: The monthly revenue R from users logging on to your gaming site depends on the monthly access fee p you charge according to the formula R( p) 5600 p2 14000 p 0 p 2.5 (R and p are in dollars.) Sketch the graph of R. Find the access fee that will result in the largest monthly revenue. Graphically Specified Function Solution: To sketch the graph of R by hand, we plot points of the form (p , R(p)) for several values of p in the domain [0 , 2.5] of R. First, we calculate several points. p 0 R(p) 0 0.5 1 1.5 2 5600 8400 8400 5600 R( p) 5600 p 14000 p 2 2.5 0 0 p 2.5 Graphically Specified Function Graphing these points gives the graph in the figure on the left, suggesting the parabola shown on the right. Graphically Specified Function The revenue graph appears to reach its highest point when p = 1.25, so setting the access fee at $1.25 appears to result in the largest monthly revenue. Graphically Specified Function Example: The following table gives the weights, in pounds, of a particular child at various ages (in months) in her first year. Age t 0 2 3 4 5 6 9 12 Weight W 8 9 13 14 16 17 18 19 Graphically Specified Function Example: The following table gives the weights, in pounds, of a particular child at various ages (in months) in her first year. Age t 0 2 3 4 5 6 9 12 Weight W 8 9 13 14 16 17 18 19 If we represent the data given in the table graphically by plotting the given pairs (t ,W(t)), we get, (connecting the successive points by line segments) W(5) = 16 W(4.5) More Examples Determine the domain, range, and intercepts of the function defined by the following graph. y 4 (2, 3) (10,0) 0 (0, -3) -4 (1, 0) (4, 0) x More Examples More Examples Consider the function f (x) x/(x+1). 1. Is the point (1,1/2) on the graph of f ? 2. If x 2, what is f (x) ? What point is on the graph of f ? 3. If f (x) 2, what is x ? What point is on the graph of f ? Average Cost Function The average cost function C of manufacturing x computers per day is given by the function 1. Determine the average cost of manufacturing 30, 40, and 50 computers per day. 2. Graph the function C(x) for 0 < x 80 and find the value of x that minimizes the average cost Finding Values of a Function or Evaluating a Function Evaluating a Function Example: Let f (x) x2 + 3x + 4. 1. Find and simplify the following. a) b) c) d) f (1), f (0), f (2) f (2x), 2 f (x) f (x + 2), f (x) + 2, f (x) + f (2) (f (x + h) f (h))/h where h0 2. Solve f (x) = 4. More Practice Problems Is the Relation a Function? Determine whether each relation represents a function. If it is a function, state the domain and range. 1. {(2, 3), (4, 1), (3, -2), (2, -1)} 2. {(-2, 3), (4, 1), (3, -2), (2, -1)} 3. {(2, 3), (4, 3), (3, 3), (2, -1)} Is the Relation a Function? 1 Determine if the equation y x 3 defines y as a 2 function of x. Determine if the equation x 2 y 2 1 defines y as a function of x. Evaluating a Function Example: Let f (x) 3x2 + 2x. 1. Find and simplify the following. a) b) c) d) f (1), f (0), f (1) f (2x), 2 f (x), f (x) f (x + 1), f (x) + 1, f (x) + f (1) (f (x + h)f (h))/h where h0 2. Solve f (x) = 5. Find the Domain of the Function Find the domain of each of the following functions. Write the answer in interval notation x4 f x 2 x 2x 3 g x x 9 2 h x 3 2x The Algebra of Functions or Sum, Difference, Product, and Quotient of Functions Function Arithmetic Suppose f and g are functions and x is in both the domain of f and the domain of g. The sum of f and g, denoted f + g, is the function defined by the formula ( f + g)(x) f(x) + g(x) The difference of f and g, denoted fg, is the function defined by the formula ( fg)(x) f(x)g(x) Function Arithmetic The sum of f and g, denoted fg, is the function defined by the formula ( f g)(x) f(x)g(x) The difference of f and g, denoted fg, is the function defined by the formula ( fg)(x) f(x)g(x) provided g(x) 0. Function Arithmetic Notice that in the previous definitions, x is in both, the domain of f and the domain of g. That is, the domain of f + g, fg, fg, and fg is the intersection of the domain of f and the domain of g. In addition, in the case of the quotient fg, the values of x that make g(x) 0 must also be excluded. Examples For f(x) 2x2 +3 and g(x) x3 +8 find f + g, fg, fg, and fg and write their domains using the interval notation. ( f + g)(x) ( fg)(x) ( f g)(x) ( fg)(x) Examples For f(x) 1/(x +2) and g(x) x /(x 1) find f + g, fg, fg, and fg and write their domains using the interval notation. ( f + g)(x) ( fg)(x) ( f g)(x) ( fg)(x)