Function notation

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Functions P 1,500 1,000 500 0 y 1970 1975 1980 1985 1990 1995 2000 2005 2010 Figure 1 Standard and Poor’s Index with dividends reinvested (credit "bull": modification of work by Prayitno hadinata; credit "graph": modification of work by measuringWorth) Introduction Toward the end of the twentieth century, the values of stocks of Internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. Figure 1 tracks the value of that initial investment of just under 100 over the 40 years. It shows that an investment that was worth less than 500 until about 1995 skyrocketed up to about 1,100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000. Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties. 159160 CHAPTER 3 f u Ncti o Ns l eARnIng Obje CTIveS In this section, you will: • Determine whether a relation represents a function. • Find the value of a function. • Determine whether a function is one-to-one. • Use the vertical line test to identify functions. • Graph the functions listed in the library of functions. 3.1 FUnCTIOnS And F UnCTIOn nOTATIOn A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships. determining Whether a Relation Represents a Function A relation is a set of ordered pairs. The set consisting of the first components of each ordered pair is called the domain and the set consisting of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first. (1, 2), (2, 4), (3, 6), (4, 8), (5, 10) e Th domain is 1, 2, 3, 4, 5. e Th range is 2, 4, 6, 8, 10. Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter x. Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter y. A function f is a relation that assigns a single element in the range to each element in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, 1, 2, 3, 4, 5, is paired with exactly one element in the range, 2, 4, 6, 8, 10. Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as (odd, 1), (even, 2), (odd, 3), (even, 4), (odd, 5) Notice that each element in the domain, even, odd is not paired with exactly one element in the range, 1, 2, 3, 4, 5. For example, the term “odd” corresponds to three values from the domain, 1, 3, 5 and the term “even” corresponds to two values from the range, 2, 4. Th is violates the definition of a function, so this relation is not a function. Figure 1 compares relations that are functions and not functions. Relation is a Function Relation is a Function Relation is NOT a Function Inputs OutputsInputs OutputsInputs Outputs x p p x p m y q q y q n z r r z (a) (b) (c) Figure 1 ( a ) This relationship is a function because each input is associated with a single output. note that input q and r both give output n. ( b ) This relationship is also a function. In this case, each input is associated with a single output. ( c ) This relationship is not a function because input q is associated with two different outputs.SECTION 3.1 f u Ncti o Ns a Nd f u Ncti o N No tioat N 161 function A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.” The input values make up the domain, and the output values make up the range. How To… Given a relationship between two quantities, determine whether the relationship is a function. 1. Identify the input values. 2. Identify the output values. 3. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function. Example 1 Determining If Menu Price Lists Are Functions e c Th oe ff e shop menu, shown in Figure 2 consists of items and their prices. a. Is price a function of the item? b. Is the item a function of the price? Menu Item Price Plain Donut .............................. 1.49 Jelly Donut .............................. 1.99 Chocolate Donut .......................... 1.99 Figure 2 Solution a. Let’s begin by considering the input as the items on the menu. The output values are then the prices. See Figure 2. Each item on the menu has only one price, so the price is a function of the item. b. Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See Figure 3. Menu Item Price Plain Donut .............................. 1.49 Jelly Donut .............................. 1.99 Chocolate Donut .......................... Figure 3 e Th refore, the item is a not a function of price. Example 2 Determining If Class Grade Rules Are Functions In a particular math class, the overall percent grade corresponds to a grade-point average. Is grade-point average a function of the percent grade? Is the percent grade a function of the grade-point average? Table 1 shows a possible rule for assigning grade points.162 CHAPTER 3 f u Ncti o Ns Percent grade 0-56 57-61 62-66 67-71 72-77 78-86 87-91 92-100 Grade-point average 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Table 1 Solution For any percent grade earned, there is an associated grade-point average, so the grade-point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specic g fi rade-point average. In the grading system given, there is a range of percent grades that correspond to the same grade-point average. For example, students who receive a grade-point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade-point average Try It 1 1 Table 2 lists the five greatest baseball players of all time in order of rank. Player Rank Babe Ruth 1 Willie Mays 2 Ty Cobb 3 Walter Johnson 4 Hank Aaron 5 Table 2 a. Is the rank a function of the player name? b. Is the player name a function of the rank? Using Function Notation Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into graphing calculators and computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions. To represent “height is a function of age,” we start by identifying the descriptive variables h for height and a for age. e l Th etters f, g, and h are often used to represent functions just as we use x, y, and z to represent numbers and A, B, and C to represent sets. h is f of a We name the function f ; height is a function of age. h = f (a) We use parentheses to indicate the function input. f (a) We name the function f ; the expression is read as “f of a.” Remember, we can use any letter to name the function; the notation h(a) shows us that h depends on a. e Th value a must be put into the function h to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication. We can also give an algebraic expression as the input to a function. For example f (a + b) means “first add a and b, and the result is the input for the function f. ” e Th operations must be performed in this order to obtain the correct result. function notation The notation y = f (x) defin es a function named f. This i s read as “y is a function of x.” The l etter x represents the input value, or independent variable. The letter y, or f (x), represents the output value, or dependent variable. 1 http://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014.SECTION 3.1 f u Ncti o Ns a Nd f u Ncti o N No tioat N 163 Example 3 Using Function Notation for Days in a Month Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Solution e n Th umber of days in a month is a function of the name of the month, so if we name the function f, we write days = f (month) or d = f (m). Th e name of the month is the input to a “rule” that associates a specic fi number (the output) with each input. 31f (January) output input rule Figure 4 For example, f (March) = 31, because March has 31 days. The notation d = f (m) reminds us that the number of days, d (the output), is dependent on the name of the month, m(the input). Analysis Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs. Example 4 Interpreting Function Notation A function N = f (y) gives the number of police oc ffi ers, N, in a town in year y. What does f (2005) = 300 represent? Solution When we read f (2005) = 300, we see that the input year is 2005. The value for the output, the number of police oc ffi ers ( N), is 300. Remember N = f (y). The s tatement f (2005) = 300 tells us that in the year 2005 there were 300 police oc ffi ers in the town. Try It 2 Use function notation to express the weight of a pig in pounds as a function of its age in days d. Q & A… Instead of a notation such as y = f (x), could we use the same symbol for the output as for the function, such as y = y (x), meaning “y is a function of x?” Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as f, which is a rule or procedure, and the output y we get by applying f to a particular input x. This is why we usually use notation such as y = f (x), P = W(d), and so on. Representing Functions Using Tables A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship. Table 3 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function f where D = f (m) identie fi s months by an integer rather than by name. Month number, m (input) 1 2 3 4 5 6 7 8 9 10 11 12 Days in month, D (output) 31 28 31 30 31 30 31 31 30 31 30 31 Table 3164 CHAPTER 3 f u Ncti o Ns Table 4 defines a function Q = g (n). Remember, this notation tells us that g is the name of the function that takes the input n and gives the output Q. n 1 2 3 4 5 Q 8 6 7 6 8 Table 4 Table 5 below displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two die ff rent output values, 40 in. and 42 in. Age in years, a (input) 5 5 6 7 8 9 10 Height in inches, h (output) 40 42 44 47 50 52 54 Table 5 How To… Given a table of input and output values, determine whether the table represents a function. 1. Identify the input and output values. 2. Check to see if each input value is paired with only one output value. If so, the table represents a function. Example 5 Identifying Tables that Represent Functions Which table, Table 6, Table 7, or Table 8, represents a function (if any)? Input Output Input Output Input Output 1 0 −3 5 2 1 5 2 0 1 5 3 5 4 4 5 8 6 Table 8 Table 6 Table 7 Solution Table 6 and Table 7 define functions. In both, each input value corresponds to exactly one output value. Table 8 does not define a function because the input value of 5 corresponds to two die ff rent output values. When a table represents a function, corresponding input and output values can also be specified using function notation. e f Th unction represented by Table 6 can be represented by writing f (2) = 1, f (5) = 3, and f (8) = 6 Similarly, the statements g (−3) = 5, g (0) = 1, and g (4) = 5 represent the function in table Table 7. Table 8 cannot be expressed in a similar way because it does not represent a function. Try It 3 Does Table 9 represent a function? Input Output 1 10 2 100 3 1000 Table 9 SECTION 3.1 f u Ncti o Ns a Nd f u Ncti o N No tioat N 165 Finding Input and Output values of a Function When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value. When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because die ff rent input values can produce the same output value. Evaluation of Functions in Algebraic Forms When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the 2 function f (x) = 5 − 3x can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5. How To… Given the formula for a function, evaluate. 1. Replace the input variable in the formula with the value provided. 2. Calculate the result. Example 6 Evaluating Functions at Specific Values 2 Evaluate f(x) = x + 3x − 4 at: f (a + h) − f (a) __ a. 2 b. a c. a + h d. h Solution Replace the x in the function with each specie fi d value. a. Because the input value is a number, 2, we can use simple algebra to simplify. 2 f (2) = 2 + 3(2) − 4 = 4 + 6 − 4 = 6 b. In this case, the input value is a letter so we cannot simplify the answer any further. 2 f (a) = a + 3a − 4 c. With an input value of a + h, we must use the distributive property. 2 f (a + h) = (a + h) + 3(a + h) − 4 2 2 = a + 2ah + h + 3a + 3h −4 d. In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that 2 2 f (a + h) = a + 2ah + h + 3a + 3h − 4 and we know that 2 f(a) = a + 3a − 4 Now we combine the results and simplify. 2 2 2 f (a + h) − f(a) (a + 2ah + h + 3a + 3h − 4) − (a + 3a − 4) __ ____ = h h 2 2ah + h + 3h ___________ = h h(2a + h + 3) ___________ = Factor out h. h = 2a + h + 3 Simplify. 166 CHAPTER 3 f u Ncti o Ns Example 7 Evaluating Functions 2 Given the function h(p) = p + 2p, evaluate h(4). Solution To evaluate h(4), we substitute the value 4 for the input variable p in the given function. 2 h(p) = p + 2p 2 h(4) = (4) +2 (4) = 16 + 8 = 24 e Th refore, for an input of 4, we have an output of 24. Try It 4 — Given the function g(m) = √ m − 4 . Evaluate g(5). Example 8 Solving Functions 2 Given the function h(p) = p + 2p, solve for h(p) = 3. Solution h(p) = 3 2 2 p + 2p = 3 Substitute the original function h(p) = p + 2p. 2 p + 2p − 3 = 0 Subtract 3 from each side. (p + 3)(p − 1) = 0 Factor. If (p + 3)(p − 1) = 0, either (p + 3) = 0 or (p − 1) = 0 (or both of them equal 0). We will set each factor equal to 0 and solve for p in each case. (p + 3) = 0, p = −3 (p − 1) = 0, p = 1 This gives us two solutions. The output h(p) = 3 when the input is either p = 1 or p = −3. We can also verify by graphing as in Figure 5. The graph verie fi s that h(1) = h(−3) = 3 and h(4) = 24. h(p) 35 30 25 20 15 10 5 p 1 2 3 4 5 p –3 –2 01 4 h(p) 30 03 24 Figure 5 Try It 5 — Given the function g(m) = √ m − 4 , solve g(m) = 2. Evaluating Functions Expressed in Formulas Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation 2n + 6p = 12 expresses a functional relationship between n and p. We can rewrite it to decide if p is a function of n.SECTION 3.1 f u Ncti o Ns a Nd f u Ncti o N No tioat N 167 How To… Given a function in equation form, write its algebraic formula. 1. Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable. 2. Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity. Example 9 Finding an Equation of a Function Express the relationship 2n + 6p = 12 as a function p = f (n), if possible. Solution To express the relationship in this form, we need to be able to write the relationship where p is a function of n, which means writing it as p = expression involving n. 2n + 6p = 12 6p = 12 − 2n Subtract 2n from both sides. 12 − 2n _______ p = Divide both sides by 6 and simplify. 6 12 2n __ ___ p = − 6 6 1 __ p = 2 − n 3 e Th refore, p as a function of n is written as 1 __ p = f (n) = 2 − n 3 Example 10 Expressing the Equation of a Circle as a Function 2 2 Does the equation x + y = 1 represent a function with x as input and y as output? If so, express the relationship as a function y = f (x). 2 Solution First we subtract x from both sides. 2 2 y = 1 − x We now try to solve for y in this equation. — 2 y = ± √1 − x — — 2 2 = + √1 − x and −√ 1 − x We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function y = f (x). If we graph both functions on a graphing calculator, we will get the upper and lower semicircles. Try It 6 3 If x − 8y = 0, express y as a function of x. Q & A… Are there relationships expressed by an equation that do represent a function but that still cannot be represented by an algebraic formula? y Yes, this can happen. For example, given the equation x = y + 2 , if we want to express y as a function of x, there is no simple algebraic formula involving only x that equals y. However, each x does determine a unique value for y, and there are mathematical procedures by which y can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for y as a function of x, even though the formula cannot be written explicitly.168 CHAPTER 3 f u Ncti o Ns Evaluating a Function Given in Tabular Form As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours. e f Th unction that relates the type of pet to the duration of its memory span is more easily visualized with the use of a 2 table. See Table 10. Pet Memory span in hours Puppy 0.008 Adult dog 0.083 Cat 16 Goldfish 2160 Beta fish 3600 Table 10 At times, evaluating a function in table form may be more useful than using equations. Here let us call the function P. The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function P at the input value of “goldfish.” We would write P(goldfish) = 2160. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function P seems ideally suited to this function, more so than writing it in paragraph or function form. How To… Given a function represented by a table, identify specic o fi utput and input values. 1. Find the given input in the row (or column) of input values. 2. Identify the corresponding output value paired with that input value. 3. F ind the given output values in the row (or column) of output values, noting every time that output value appears. 4. Identify the input value(s) corresponding to the given output value. Example 11 Evaluating and Solving a Tabular Function Using Table 11, a. Evaluate g(3) b. Solve g(n) = 6. n 1 2 3 4 5 g (n) 8 6 7 6 8 Table 11 Solution a. Evaluating g (3) means determining the output value of the function g for the input value of n = 3. The table output value corresponding to n = 3 is 7, so g (3) = 7. b. S olving g (n) = 6 means identifying the input values, n, that produce an output value of 6. Table 11 shows two solutions: 2 and 4. When we input 2 into the function g, our output is 6. When we input 4 into the function g, our output is also 6. 2 http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.SECTION 3.1 f u Ncti o Ns a Nd f u Ncti o N No tioat N 169 Try It 7 Using Table 11, evaluate g(1). Finding Function Values from a Graph Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value( s ). Example 12 Reading Function Values from a Graph Given the graph in Figure 6, f (x) a. Evaluate f (2). b. Solve f (x) = 4. 7 6 5 4 3 2 1 –5–4 –3 –2 –1 14 2 3 5 –1 2 – –3 Figure 6 Solution a. T o evaluate f (2), locate the point on the curve where x = 2, then read the y-coordinate of that point. The point has coordinates (2, 1), so f (2) = 1. See Figure 7. f (x) 7 6 5 4 3 (2, 1) 2 f (2) = 1 1 –5–4 –3 –2 –1 14 2 3 5 –1 2 – –3 Figure 7 b. To solve f (x) = 4, we find t he output value 4 on the vertical axis. Moving horizontally along the line y = 4, we locate two points of the curve with output value 4: (−1, 4) and (3, 4). These p oints represent the two solutions to f (x) = 4: −1 or 3. This means f (−1) = 4 and f (3) = 4, or when the input is −1 or 3, the output is 4. See Figure 8. f (x) 7 6 5 4 (−1, 4) (3, 4) 3 2 1 –5–4 –3 –2 –1 14 2 3 5 –1 2 – –3 Figure 8170 CHAPTER 3 f u Ncti o Ns Try It 8 Using Figure 7, solve f (x) = 1. determining Whether a Function is One-to-One Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in Figure 1 at the beginning of this chapter, the stock price was 1,000 on five die ff rent dates, meaning that there were five die ff rent input values that all resulted in the same output value of 1,000. However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in Table 12. Letter grade Grade-point average A 4.0 B 3.0 C 2.0 D 1.0 Table 12 This grading system represents a one-to-one function, because each letter input yields one particular grade-point average output and each grade-point average corresponds to one input letter. To visualize this concept, let’s look again at the two simple functions sketched in Figure 1(a) and Figure 1(b). The function in part ( a) shows a relationship that is not a one-to-one function because inputs q and r both give output n. The function in part (b ) shows a relationship that is a one-to-one function because each input is associated with a single output. one-to-one function A one-to-one function is a function in which each output value corresponds to exactly one input value. There are no repeated x- or y-values. Example 13 Determining Whether a Relationship Is a One-to-One Function Is the area of a circle a function of its radius? If yes, is the function one-to-one? 2 Solution A circle of radius r has a unique area measure given by A = πr , so for any input, r, there is only one output, A. The area is a function of radius r. If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area 2 measure A is given by the formula A = πr . Because areas and radii are positive numbers, there is exactly one solution: ___ A __ r = So the area of a circle is a one-to-one function of the circle’s radius. √ π Try It 9 a. Is a balance a function of the bank account number? b. Is a bank account number a function of the balance? c. Is a balance a one-to-one function of the bank account number? Try It 10 a. I f each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade? b. If so, is the function one-to-one?SECTION 3.1 f u Ncti o Ns a Nd f u Ncti o N No tioat N 171 Using the vertical l ine Test As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide oe ft n makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis. e m Th ost common graphs name the input value x and the output value y, and we say y is a function of x, or y = f (x) when the function is named f. The graph of the function is the set of all points ( x, y) in the plane that satisfies the equation y = f (x). If the function is defined for only a few input values, then the graph of the function consists of only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 9 tell us that f (0) = 2 and f (6) = 1. However, the set of all points (x, y) satisfying y = f (x) is a curve. e Th curve shown includes (0, 2) and (6, 1) because the curve passes through those points. y x Figure 9 The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See Figure 10. Function Not a Function Not a Function Figure 10 How To… Given a graph, use the vertical line test to determine if the graph represents a function. 1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once. 2. If there is any such line, determine that the graph does not represent a function.172 CHAPTER 3 f u Ncti o Ns Example 14 Applying the Vertical Line Test Which of the graphs in Figure 11 represent( s ) a function y = f (x)? y f (x) f (x) 5 4 3 2 1 x x x 4 5 6 7 8 9 10 11 12 (a) (b) (c) Figure 11 Solution If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts ( a) and (b) of Figure 11. From this we can conclude that these two graphs represent functions. e Th third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point, as shown in Figure 12. y x Figure 12 Try It 11 Does the graph in Figure 13 represent a function? y x Figure 13SECTION 3.1 f u Ncti o Ns a Nd f u Ncti o N No tioat N 173 Using the horizontal l ine Test Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function. How To… Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function. 1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once. 2. If there is any such line, determine that the function is not one-to-one. Applying the Horizontal Line Test Example 15 Consider the functions shown in Figure 11(a) and Figure 11(b). Are either of the functions one-to-one? Solution e Th function in Figure 11(a) is not one-to-one. The horizontal line shown in Figure 14 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.) f (x) x Figure 14 e f Th unction in Figure 11(b) is one-to-one. Any horizontal line will intersect a diagonal line at most once. Try It 12 y Is the graph shown here one-to-one? x Identifying basic Toolkit Functions In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which 174 CHAPTER 3 f u Ncti o Ns we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use x as the input variable and y = f (x) as the output variable. We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in Table 13. Toolkit Functions Name Function Graph f (x) x f(x) –2 2 x Constant f (x) = c, where c is a constant 02 22 f (x) f(x) x –2 –2 x Identity f (x) = x 00 22 f (x) f(x) x –2 2 x Absolute value f (x) =  x ∣ ∣ 00 22 f (x) x f(x) –2 4 –1 1 2 x Quadratic f (x) = x 00 1 1 2 4 f (x) x f(x) –1 –1 –0.5 –0.125 3 x Cubic f (x) = x 00 0.5 0.125 1 1SECTION 3.1 f u Ncti o Ns a Nd f u Ncti o N No tioat N 175 f (x) x f(x) –2 –0.5 –1 –1 1 –0.5 –2 __ x Reciprocal f (x) = x 0.5 2 11 2 0.5 f (x) x f(x) –2 0.25 –1 1 1 _ –0.5 4 x Reciprocal squared f (x) = 2 x 0.5 4 11 2 0.25 f (x) f(x) x — 00 x Square root f (x) = √ x 11 42 f (x) x f(x) –1 –1 –0.125 –0.5 3 — Cube root x f (x) = √ x 00 0.125 0.5 1 1 Table 13 Access the following online resources for additional instruction and practice with functions. • determine if a Relation is a Function (http://openstaxcollege.org/l/relationfunction) • vertical l ine Test (http://openstaxcollege.org/l/vertlinetest) • Introduction to Functions (http://openstaxcollege.org/l/introtofunction) • vertical l ine Test of graph (http://openstaxcollege.org/l/vertlinegraph) • One-to-one Functions (http://openstaxcollege.org/l/onetoone) • graphs as One-to-one Functions (http://openstaxcollege.org/l/graphonetoone)176 CHAPTER 3 f u Ncti o Ns 3.1 SeCTIOn exeRCISeS veRbAl 1. What is the difference between a relation and 2. What is the difference between the input and the a function? output of a function? 3. Why does the vertical line test tell us whether the 4. How can you determine if a relation is a one-to-one graph of a relation represents a function? function? 5. Why does the horizontal line test tell us whether the graph of a function is one-to-one? Algeb RAIC For the following exercises, determine whether the relation represents a function. 6. (a, b), (c, d), (a, c) 7. (a, b),(b, c),(c, c) For the following exercises, determine whether the relation represents y as a function of x. 2 2 8. 5x + 2y = 10 9. y = x 10. x = y 2 2 2 11. 3x + y = 14 12. 2x + y = 6 13. y = −2x + 40x 3y + 5 — 1 __ _ 2 14. y = 15. x = 16. x = √ 1 − y x 7y − 1 2 2 3x + 5 ______ 18. x + y = 9 19. 2xy = 1 17. y = 7x − 1 3 3 — 20. x = y 21. y = x 2 22. y = √1 − x — — 2 2 25. y = x 23. x = ±√ 1 − y 24. y = ± √1 − x 3 2 26. y = x For the following exercises, evaluate the function f at the indicated values f (−3), f (2), f (−a), −f (a), f (a + h). — 2 27. f (x) = 2x − 5 28. f (x) = −5x + 2x − 1 29. f (x) = √ 2 − x + 5 6x − 1 ______ 31. f (x) =  x − 1 −  x + 1 ∣ ∣ 30. f (x) = ∣ ∣ 5x + 2 g(x + h) − g(x) 2 __ 32. Given the function g(x) = 5 − x , simplify , h ≠ 0 h g(x) − g(a) 2 _ 33. Given the function g(x) = x + 2x, simplify , x ≠ a x − a 34. Given the function k(t) = 2t − 1: 35. Given the function f (x) = 8 − 3x: a. Evaluate k(2). a. Evaluate f (−2). b. Solve k(t) = 7. b. Solve f (x) = −1. 2 2 36. Given the function p(c) = c + c: 37. Given the function f (x) = x − 3x a. Evaluate p(−3). a. Evaluate f (5). b. Solve p(c) = 2. b. Solve f (x) = 4 — 39. Consider the relationship 3r + 2t = 18. 38. Given the function f (x) = √ x + 2 : a. Write the relationship as a function r = f (t). a. Evaluate f (7). b. Evaluate f (−3). b. Solve f (x) = 4 c. Solve f (t) = 2. SECTION 3.1 s ectio N e xercises 177 gRAPhICAl For the following exercises, use the vertical line test to determine which graphs show relations that are functions. y y y 40. 41. 42. x x x y y 44. 43. y 45. x x x y y y 46. 47. 48. x x x y y y 49. 50. 51. x x x178 CHAPTER 3 f u Ncti o Ns 52. Given the following graph 53. Given the following graph 54. Given the following graph a. Evaluate f (−1). a. Evaluate f (0). a. Evaluate f (4). b. Solve for f (x) = 3. b. Solve for f (x) = −3. b. Solve for f (x) = 1. y y y x x x For the following exercises, determine if the given graph is a one-to-one function. y y y 55. 56. 57. x x x y y 58. 59. 5 4 3 2 1 x x π π 1 2 3 4 5 nUmeRIC For the following exercises, determine whether the relation represents a function. 60. (−1, −1),(−2, −2),(−3, −3) 61. (3, 4),(4, 5),(5, 6) 62. (2, 5),(7, 11),(15, 8),(7, 9) For the following exercises, determine if the relation represented in table form represents y as a function of x. 64. 65. 63. x 5 10 10 x 5 10 15 x 5 10 15 y 3 8 14 y 3 8 14 y 3 8 8 For the following exercises, use the function f represented in Table 14 below. x 0 1 2 3 4 5 6 7 8 9 f (x) 74 28 1 53 56 3 36 45 14 47 Table 14 67. Solve f (x) = 1 66. Evaluate f (3).