Domain of Rational Functions

Transcription

1 SECTION 46 RATIONAL FU NCTIONS SKI LLS OBJ ECTIVES Find the domain of a rational function Determine vertical, horizontal, and slant asmptotes of rational functions Graph rational functions CONCE PTUAL OBJ ECTIVES Understand arrow notation Interpret the behavior of the graph of a rational function near an asmptote Domain of Rational Functions So far in this chapter we have discussed polnomial functions We now turn our attention to rational functions, which are ratios of polnomial functions Ratios of integers are called rational numbers Similarl, ratios of polnomial functions are called rational functions D E F I N I T I O N Rational Function A function f() is a rational function if f() = n() d() d() Z 0 where the numerator, n(), and the denominator, d(), are polnomial functions The domain of f () is the set of all real numbers such that d() Z 0 Note: If d() is a constant, then f() is a polnomial function The domain of an polnomial function is the set of all real numbers When we divide two polnomial functions, the result is a rational function, and we must eclude an values of that make the denominator equal to zero 44

2 446 CHAPTE R 4 Polnomial and Rational Functions EXAM PLE Finding the Domain of a Rational Function + Find the domain of the rational function f() = Epress the domain in interval notation Set the denominator equal to zero Factor ( 2)( 3) 0 Solve for 2 or 3 Eliminate these values from the domain Z -2 or Z 3 State the domain in interval notation (-, -2) (-2, 3) (3, ) Answer: The domain is the set of all real numbers such that Z - or Z 4 Interval notation: (-, -) (-, 4) (4, ) YOU R TU R N Find the domain of the rational function f() = Epress the domain in interval notation It is important to note that there are not alwas restrictions on the domain For eample, if the denominator is never equal to zero, the domain is the set of all real numbers EXAM PLE 2 When the Domain of a Rational Function Is the Set of All Real Numbers 3 Find the domain of the rational function g() = Epress the domain in interval notation Set the denominator equal to zero Subtract 9 from both sides Solve for There are no real solutions; therefore the domain has no restrictions State the domain in interval notation 2 9 3i or 3i R, the set of all real numbers (-, ) Answer: The domain is the set of all real numbers Interval notation: (, ) YO U R T U R N Find the domain of the rational function g() = domain in interval notation Epress the hole It is important to note that f() = 2-4, where Z -2, and g() = - 2 are not + 2 the same function Although f() can be written in the factored form ( - 2)( + 2) f() = = - 2, its domain is different The domain of g() is the set of + 2 all real numbers, whereas the domain of f() is the set of all real numbers such that Z -2 If we were to plot f() and g(), the would both look like the line 2 However, f() would have a hole, or discontinuit, at the point 2

3 46 Rational Functions 447 Vertical, Horizontal, and Slant Asmptotes If a function is not defined at a point, then it is still useful to know how the function behaves near that point Let s start with a simple rational function, the reciprocal function f() = This function is defined everwhere ecept at 0 f () = (, ) f() undefined approaching 0 from the left approaching 0 from the right 0 0 f () (, ) We cannot let 0 because that point is not in the domain of the function We should, however, ask the question, how does f() behave as approaches zero? Let us take values that get closer and closer to 0, such as 0, 00, 000, (See the table above) We use an arrow to represent the word approach, a positive superscript to represent from the right, and a negative superscript to represent from the left A plot of this function can be generated using point-plotting techniques The following are observations of the graph f() = WORDS As approaches zero from the right, the function f() increases without bound As approaches zero from the left, the function f() decreases without bound As approaches infinit (increases without bound), the function f() approaches zero from above As approaches negative infinit (decreases without bound), the function f() approaches zero from below MATH S 0 S ˆ S 0 S ˆ S ˆ S 0 S ˆ S 0 The smbol does not represent an actual real number This smbol represents growing without bound f () = (, ) (, ) Notice that the function is not defined at 0 The -ais, or the vertical line 0, represents the vertical asmptote 2 Notice that the value of the function is never equal to zero The -ais is never touched b the function The -ais, or 0, is a horizontal asmptote

4 448 CHAPTE R 4 Polnomial and Rational Functions Asmptotes are lines that the graph of a function approaches Suppose a football team s defense is its own 8 ard line and the team gets an offsides penalt that results in loss of half the distance to the goal Then the offense would get the ball on the 4 ard line Suppose the defense gets another penalt on the net pla that results in half the distance to the goal The offense would then get the ball on the 2 ard line If the defense received 0 more penalties all resulting in half the distance to the goal, would the referees give the offense a touchdown? No, because although the offense ma appear to be snapping the ball from the goal line, technicall it has not actuall reached the goal line Asmptotes utilize the same concept We will start with vertical asmptotes Although the function f() = had one vertical asmptote, in general, rational functions can have none, one, or several vertical asmptotes We will first formall define what a vertical asmptote is and then discuss how to find it D E F I N I T I O N Vertical Asmptotes The line a is a vertical asmptote for the graph of a function if f() either increases or decreases without bound as approaches a from either the left or the right = a = a = a f () f () = a f () f () Vertical asmptotes assist us in graphing rational functions since the essentiall steer the function in the vertical direction How do we locate the vertical asmptotes of a rational function? Set the denominator equal to zero If the numerator and denominator have no common factors, then an numbers that are ecluded from the domain of a rational function locate vertical asmptotes A rational function f() = n() is said to be in lowest terms if the numerator n() and d() denominator d() have no common factors Let f() = n() be a rational function in lowest d() terms; then an zeros of the numerator n() correspond to -intercepts of the graph of f, and an zeros of the denominator d() correspond to vertical asmptotes of the graph of f If a rational function does have a common factor (is not in lowest terms), then the common factor(s) should be canceled, resulting in an equivalent rational function R() in lowest

5 46 Rational Functions 449 terms If ( a) p is a factor of the numerator and ( a) q is a factor of the denominator, then there is a hole in the graph at a provided p q and a is a vertical asmptote if p q LOCATING VE RTICAL ASYM PTOTES Let f() = n() be a rational function in lowest terms (that is, assume n() and d() are d() polnomials with no common factors); then the graph of f has a vertical asmptote at an real zero of the denominator d() That is, if d(a) 0, then a corresponds to a vertical asmptote on the graph of f Note: If f is a rational function that is not in lowest terms, then divide out the common factors, resulting in a rational function R that is in lowest terms An common factor a of the function f corresponds to a hole in the graph of f at a provided the multiplicit of a in the numerator is greater than or equal to the multiplicit of a in the denominator Stud Tip The vertical asmptotes of a rational function in lowest terms occur at -values that make the denominator equal to zero EXAM PLE 3 Determining Vertical Asmptotes Locate an vertical asmptotes of the rational function f() = Factor the denominator The numerator and denominator have no common factors Set the denominator equal to zero f() = 2 + = (2 + )(3-2) and 3-2 = 0 Solve for = - 2 and = 2 3 The vertical asmptotes are = - and = YOU R TU R N Locate an vertical asmptotes of the following rational function: Answer: = - 2 and 3 f() =

6 40 CHAPTE R 4 Polnomial and Rational Functions E X A M P L E 4 Determining Vertical Asmptotes When the Rational Function Is Not in Lowest Terms Locate an vertical asmptotes of the rational function f() = Factor the denominator ( 2 3 0) ( )( 2) Write the rational function in factored form ( + 2) f() = ( - )( + 2) Cancel (divide out) the common factor ( 2) R() = ( - ) Z -2 Find the values when the denominator of R is equal to zero 0 and The vertical asmptotes are = 0 and = Note: 2 is not in the domain of f(), even though there is no vertical asmptote there There is a hole in the graph at 2 Graphing calculators do not alwas show such holes Answer: 3 YOU R TU R N Locate an vertical asmptotes of the following rational function: We now turn our attention to horizontal asmptotes As we have seen, rational functions can have several vertical asmptotes However, rational functions can have at most one horizontal asmptote Horizontal asmptotes impl that a function approaches a constant value as becomes large in the positive or negative direction Another difference between vertical and horizontal asmptotes is that the graph of a function never touches a vertical asmptote but, as ou will see in the net bo, the graph of a function ma cross a horizontal asmptote, just not at the ends ( S ; ) D E F I N I T I O N Horizontal Asmptote The line b is a horizontal asmptote of the graph of a function if f() approaches b as increases or decreases without bound The following are three eamples: f () f() = As S, f() S b f () = b = b = b f () Note: A horizontal asmptote steers a function as gets large Therefore, when is not large, the function ma cross the asmptote

7 46 Rational Functions 4 How do we determine whether a horizontal asmptote eists? And, if it does, how do we locate it? We investigate the value of the rational function as S or as S - One of two things will happen: either the rational function will increase or decrease without bound or the rational function will approach a constant value We sa that a rational function is proper if the degree of the numerator is less than the degree of the denominator Proper rational functions, like f() =, approach zero as gets large Therefore, all proper rational functions have the specific horizontal asmptote, 0 (see Eample a) We sa that a rational function is improper if the degree of the numerator is greater than or equal to the degree of the denominator In this case, we can divide the numerator b the denominator and determine how the quotient behaves as increases without bound If the quotient is a constant (resulting when the degrees of the numerator and denominator are equal), then as S or as S -, the rational function approaches the constant quotient (see Eample b) If the quotient is a polnomial function of degree or higher, then the quotient depends on and does not approach a constant value as increases (see Eample c) In this case, we sa that there is no horizontal asmptote We find horizontal asmptotes b comparing the degree of the numerator and the degree of the denominator There are three cases to consider: The degree of the numerator is less than the degree of the denominator 2 The degree of the numerator is equal to the degree of the denominator 3 The degree of the numerator is greater than the degree of the denominator LOCATING HOR IZONTAL ASYM PTOTES Let f be a rational function given b f() = n() d() = a n n + a n- n- + Á + a + a 0 b m m + b m- m- + Á + b + b 0 where n() and d() are polnomials When n m, the -ais ( 0) is the horizontal asmptote 2 When n m, the line = a n (ratio of leading coefficients) is the horizontal asmptote 3 When n m, there is no horizontal asmptote b m In other words, When the degree of the numerator is less than the degree of the denominator, then 0 is the horizontal asmptote 2 When the degree of the numerator is the same as the degree of the denominator, then the horizontal asmptote is the ratio of the leading coefficients 3 If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asmptote

8 42 CHAPTE R 4 Polnomial and Rational Functions Technolog Tip The following graphs correspond to the rational functions given in Eample The horizontal asmptotes are apparent, but are not drawn in the graph a Graph f() = b Graph g() = c Graph h() = EXAM PLE Finding Horizontal Asmptotes Determine whether a horizontal asmptote eists for the graph of each of the given rational functions If it does, locate the horizontal asmptote a b g() = f() = c Solution (a): The degree of the numerator 8 3 is n The degree of the denominator 4 2 is 2 m 2 The degree of the numerator is less than the degree of the denominator n m The -ais is the horizontal asmptote for the graph of f() 0 The line Solution (b): is the horizontal asmptote for the graph of f() The degree of the numerator is 2 n 2 The degree of the denominator 4 2 is 2 m 2 The degree of the numerator is equal to the degree of the denominator The ratio of the leading coefficients is the horizontal asmptote for the graph of g() The line is the horizontal asmptote for the graph of g() If we divide the numerator b the denominator, the resulting quotient is the constant 2 Solution (c): = 0 = 2 n m The degree of the numerator is 3 n 3 The degree of the denominator 4 2 is 2 m 2 The degree of the numerator is greater than the degree of the denominator The graph of the rational function h() has no horizontal asmptote h() = = 8 4 = 2 g() = = n m If we divide the numerator b the denominator, the resulting quotient is a linear function h() = = Answer: = is the horizontal asmptote YOU R TU R N Find the horizontal asmptote (if one eists) for the graph of the rational function f() = Stud Tip There are three tpes of linear asmptotes: horizontal, vertical, and slant Thus far we have discussed linear asmptotes: vertical and horizontal There are three tpes of lines: horizontal (slope is zero), vertical (slope is undefined), and slant (nonzero slope) Similarl, there are three tpes of linear asmptotes: horizontal, vertical, and slant

9 46 Rational Functions 43 Recall that when dividing polnomials the degree of the quotient is alwas the difference between the degree of the numerator and the degree of the denominator For eample, a cubic (third-degree) polnomial divided b a quadratic (second-degree) polnomial results in a linear (first-degree) polnomial A fifth-degree polnomial divided b a fourth-degree polnomial results in a first-degree (linear) polnomial When the degree of the numerator is eactl one more than the degree of the denominator, the quotient is linear and represents a slant asmptote SLANT ASYM PTOTES Let f be a rational function given b f() = n(), where n() and d() are polnomials d() and the degree of n() is one more than the degree of d() On dividing n() b d(), the rational function can be epressed as f() = m + b + r() d() where the degree of the remainder r() is less than the degree of d() and the line m b is a slant asmptote for the graph of f Note that as S - or S, f () S m + b EXAM PLE 6 Finding Slant Asmptotes Determine the slant asmptote of the rational function f() = Divide the numerator b the denominator with long division Note that as S ; the rational epression approaches 0 The quotient is the slant asmptote ( ) ( ) f() = S 0 as S ; = 4 + Technolog Tip The graph of f() = has a slant asmptote of = 4 + YO U R T U R N Find the slant asmptote of the rational function f() = Answer:

10 44 CHAPTE R 4 Polnomial and Rational Functions Stud Tip Common factors need to be divided out first; then the remaining -values corresponding to a denominator value of 0 are vertical asmptotes Graphing Rational Functions We can now graph rational functions using asmptotes as graphing aids The following bo summarizes the five-step procedure for graphing rational functions GRAPHING RATIONAL FUNCTIONS Let f be a rational function given b f() = n() d() Step : Find the domain of the rational function f Step 2: Find the intercept(s) -intercept: evaluate f(0) -intercept: solve the equation n() 0 for in the domain of f Step 3: Find an holes Factor the numerator and denominator Divide out common factors A common factor a corresponds to a hole in the graph of f at a if the multiplicit of a in the numerator is greater than or equal to the multiplicit of a in the denominator The result is an equivalent rational function R() = p() in lowest terms q() Step 4: Find an asmptotes Vertical asmptotes: solve q() 0 Compare the degree of the numerator and the degree of the denominator to determine whether either a horizontal or slant asmptote eists If one eists, find it Step : Find additional points on the graph of f particularl near asmptotes Step 6: Sketch the graph; draw the asmptotes, label the intercept(s) and additional points, and complete the graph with a smooth curve between and beond the vertical asmptotes It is important to note that an real number eliminated from the domain of a rational function corresponds to either a vertical asmptote or a hole on its graph Stud Tip An real number ecluded from the domain of a rational function corresponds to either a vertical asmptote or a hole on its graph EXAM PLE 7 Graphing a Rational Function Graph the rational function f() = 2-4 STEP Find the domain Set the denominator equal to zero Solve for ;2 State the domain STEP 2 Find the intercepts -intercept: f(0) = = 0 -intercepts: f() = = 0 The onl intercept is at the point (0, 0) (-, -2) (-2, 2) (2, )

11 46 Rational Functions 4 STEP 3 Find an holes There are no common factors, so f is in lowest terms Since there are no common factors, there are no holes on the graph of f STEP 4 Find an asmptotes f() = Vertical asmptotes: d() ( 2)( 2) 0 = -2 ( + 2)( - 2) and = 2 Horizontal asmptote: Degree of numerator = Degree of denominator = 2 Degree of numerator Degree of denominator 0 STEP Find additional points on the graph 3 3 f() STEP 6 Sketch the graph; label the intercepts, asmptotes, and additional points and complete with a smooth curve approaching the asmptotes Answer: YOU R TU R N Graph the rational function f() = 2 - E X A M P L E 8 Graphing a Rational Function with No Horizontal or Slant Asmptotes State the asmptotes (if there are an) and graph the rational function f() = STEP Find the domain Set the denominator equal to zero 2 0 Solve for ; State the domain STEP 2 Find the intercepts -intercept: -intercepts: n() Factor 2 ( 3)( 2) 0 Solve The intercepts are the points (0, 0), (3, 0), and ( 2, 0) (-, -) (-, ) (, ) f(0) = 0 - = 0 0, 3, and 2 Technolog Tip The behavior of each function as approaches or - can be shown using tables of values Graph f () =

12 46 CHAPTE R 4 Polnomial and Rational Functions The graph of f() shows that the vertical asmptotes are at = ; and there is no horizontal asmptote or slant asmptote STEP 3 Find an holes There are no common factors, so f is in lowest terms Since there are no common factors, there are no holes on the graph of f STEP 4 Find the asmptotes Vertical asmptote: d() 2 0 Factor ( )( ) 0 Solve and No horizontal asmptote: degree of n() degree of d() [4 2] No slant asmptote: degree of n() degree of d() [4 2 2 ] The asmptotes are and STEP Find additional points on the graph f() = 2 ( - 3)( + 2) ( - )( + ) f() Answer: Vertical asmptote: 2 No horizontal or slant asmptotes 0 STEP 6 Sketch the graph; label the intercepts and asmptotes, and complete with a smooth curve between and beond the vertical asmptote 0 = ( 2, 0) (3, 0) = = 2 0 (, 0) (3, 0) 0 2 YOU R TU R N State the asmptotes (if there are an) and graph the rational function f() = EXAM PLE 9 Graphing a Rational Function with a Horizontal Asmptote State the asmptotes (if there are an) and graph the rational function STEP Find the domain Set the denominator equal to zero Solve for 2 State the domain f() = (-, 2) (2, )

13 46 Rational Functions 47 STEP 2 Find the intercepts -intercept: -intercepts: f(0) = 0 8 = 0 n() Factor 2(2 )( 3) 0 Solve 0, = and 3 2, Technolog Tip The behavior of each function as approaches or - can be shown using tables of values Graph f () = The intercepts are the points (0, 0), A 2, 0B, and ( 3, 0) STEP 3 Find the holes f() = 2(2 - )( + 3) (2 - )( ) There are no common factors, so f is in lowest terms (no holes) STEP 4 Find the asmptotes Vertical asmptote: d() Solve 2 Horizontal asmptote: Use leading coefficients The asmptotes are 2 and 4 degree of n() degree of d() = 4 - = -4 The graph of f() shows that the vertical asmptote is at = 2 and the horizontal asmptote is at = -4 STEP Find additional points on the graph 4 3 f() STEP 6 Sketch the graph; label the intercepts and asmptotes and complete with a smooth curve 0 ( 3, 0) = (, 0) 2 = 4 0 Answer: Vertical asmptotes: 4, Horizontal asmptote: 2 Intercepts: A0, B, A3 2, 0B, (2, 0) = 2 YOU R TU R N Graph the rational function f() = Give equations of the vertical and horizontal asmptotes and state the intercepts 4 6 = = 4

14 48 CHAPTE R 4 Polnomial and Rational Functions Technolog Tip The behavior of each function as approaches or - can be shown using tables of values Graph f() = EXAM PLE 0 Graph the rational function STEP Find the domain Graphing a Rational Function with a Slant Asmptote Set the denominator equal to zero 2 0 Solve for State the domain STEP 2 Find the intercepts f() = (-,-2) (-2, ) -intercept: f(0) = = -2 The graph of f() shows that the vertical asmptote is at = -2 and the slant asmptote is at = - -intercepts: n() Factor ( )( 4) 0 Solve and 4 The intercepts are the points (0, 2), (, 0), and (4, 0) STEP 3 Find an holes There are no common factors, so f is in lowest terms Since there are no common factors, there are no holes on the graph of f STEP 4 Find the asmptotes Vertical asmptote: d() 2 0 Solve 2 Slant asmptote: degree of n() degree of d() Divide n() b d() Write the equation of the asmptote The asmptotes are 2 and STEP Find additional points on the graph ( - 4)( + ) f() = ( + 2) f() = = f() Answer: Horizontal asmptote: 3 Slant asmptote: 4 20 STEP 6 Sketch the graph; label the intercepts and asmptotes, and complete with a smooth curve between and beond the vertical asmptote 20 = 2 (, 0) 20 (0, 2) (4, 0) 20 = ( 0, ) 2 3 = + 4 (, 0) 0 0 ( 2, 0) 0 = 3 20 YOU R TU R N For the function f() = , state the asmptotes (if an eist) - 3 and graph the function

15 46 Rational Functions 49 EXAM PLE Graph the rational function f() = STEP Find the domain Graphing a Rational Function with a Hole in the Graph Set the denominator equal to zero Solve for ( 2)( ) 0 State the domain STEP 2 Find the intercepts or 2 -intercept: f(0) = = 3 -intercepts: n() ( 3)( 2) 0 3 or 2 The intercepts correspond to the points (0, 3) and (-3, 0) The point (2, 0) appears to be an -intercept; however, 2 is not in the domain of the function STEP 3 Find an holes Since 2 is a common factor, there is a hole in the graph of f at 2 Dividing out the common factor generates an equivalent rational function in lowest terms STEP 4 Find the asmptotes Vertical asmptotes: 0 Horizontal asmptote: (-, -) (-, 2) (2, ) ( - 2)( + 3) f() = ( - 2)( + ) ( + 3) R() = ( + ) = - Degree of numerator of f Degree of denominator of f 2 and Degree of numerator of R Degree of denominator of R Technolog Tip The behavior of each function as approaches or - can be shown using tables of values Graph f() = The graph of f() shows that the vertical asmptote is at = - and the horizontal asmptote is at = Notice that the hole at = 2 is not apparent in the graph A table of values supports the graph Since the degree of the numerator equals the degree of the denominator, use the leading coefficients = = STEP Find additional points on the graph f() or R() STEP 6 Sketch the graph; label the intercepts, asmptotes, and additional points and complete with a smooth curve approaching asmptotes Recall the hole at = 2 Note that R(2) = 3 so the open hole is located at the point (2, /3) Answer: 7 3 YOU R TU R N Graph the rational function f() =

16 460 CHAPTE R 4 Polnomial and Rational Functions SECTION 46 SU M MARY In this section, rational functions were discussed f() = n() d() Domain: All real numbers ecept the -values that make the denominator equal to zero, d() 0 Vertical Asmptotes: Vertical lines, a, where d(a) a, after all common factors have been divided out Vertical asmptotes steer the graph and are never touched Horizontal Asmptotes: Horizontal lines, b, that steer the graph as S ; If degree of the numerator degree of the denominator, then 0 is a horizontal asmptote 2 If degree of the numerator degree of the denominator, then c is a horizontal asmptote where c is the ratio of the leading coefficients of the numerator and denominator, respectivel 3 If degree of the numerator degree of the denominator, then there is no horizontal asmptote Slant Asmptotes: Slant lines, m b, that steer the graph as S ; If degree of the numerator degree of the denominator, then there is a slant asmptote 2 Divide the numerator b the denominator The quotient corresponds to the equation of the line (slant asmptote) Procedure for Graphing Rational Functions Find the domain of the function 2 Find the intercept(s) -intercept -intercepts (if an) 3 Find an holes If a is a common factor of the numerator and denominator, then a corresponds to a hole in the graph of the rational function if the multiplicit of a in the numerator is greater than or equal to the multiplicit of a in the denominator The result after the common factor is canceled is an equivalent rational function in lowest terms (no common factor) 4 Find an asmptotes Vertical asmptotes Horizontal/slant asmptotes Find additional points on the graph 6 Sketch the graph: draw the asmptotes and label the intercepts and points and connect with a smooth curve SECTION 46 EXE RCISES SKILLS In Eercises 0, find the domain of each rational function f() = 2 f() = 3 f() = (3 + )(2 - ) f() = 6 f() = 7 f() = f() = - 3( ) 0 2( ) f() = ( ) ( ) f() = - 3 (2-3)( - 7) f() = In Eercises 20, find all vertical asmptotes and horizontal asmptotes (if there are an) 2 3 f() = 73 + f() = f() = f() = f() = f() = (02-3)(2 + 4) 9 20 f() = f() = 07( - 0)( ) 2-02 f() = f() = 0 A B 2 -

17 46 Rational Functions 46 In Eercises 2 26, find the slant asmptote corresponding to the graph of each rational function 2 22 f() = f() = f() = f() = In Eercises 27 32, match the function to the graph f() = 28 f() = f() = f() = a b c 0 0 f() = f() = f() = f() = d e f In Eercises 33 8, use the graphing strateg outlined in this section to graph the rational functions f() = 34 f() = 3 f() = 36 f() = 37 f() = f() = 3(2 - ) f() = 2( ) 4 f() = f() = 2 + f() = f() = f() = 48 f() = 49 f() = (2 + ) 2 (3 + ) 4 ( ) 3 ( - )2 f() = f() = f() = 4 f() = ( 2 - ) 3( - ) 6 f() = ( - )(2-9) f() = ( - )(2-4) 7 f() = 8 f() = ( - 2)( 2 + ) ( - 3)( 2 + ) ( 2-4) f() = f() = f () = 22 - (6 2 - ) 2 ( + )2 ( 2 - ) f() = ( - 3) ( 2 + )