3 6 x a. 12 b. 63 c. 27 d. 0. 6, find

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1 Advanced Algebra Topics COMPASS Review revised Summer 0 You will be allowed to use a calculator on the COMPASS test Acceptable calculators are basic calculators, scientific calculators, and approved models of graphing calculators For more information, see the JCCC Testing Services website at or call If 6 f ( ) and g( ) ( )( ), find f() g( ) a b 6 c d 0 If a f ( ), g( ), and 5 b, find f ( ) g( ) h( ) h( ) 5 5 c 5 d 5 If f ( ) and a g( ) 8 9 6, find b g( ) f ( ) c 9 d If f ( ) and a b g( ), find ( g f )( ) c 9 d 5 If a f ( ), find f( ) b c d 0 6 If f ( ) ( ), find f ( h) f ( ) a 6h h h b h c h h h d 6 If f ( ) and g( ), find f[ g ()] a b c d 8 If f ( ), find f a f ( ) ( ) b f ( ) c f ( ) d f ( ) 9 If f ( ) 6, and g is a function such that f ( g( )) g( f ( )) for all values of, find g ( ) a g( ) 6 b g( ) c g( ) 6 d g( ) If f( ) contains the point (,), then f ( ) must contain what point? a (,) b (,) c (, ) d (, ) The value V in dollars of a piece of equipment t months after its purchase is given by the function V( t) 50 5t Find V (50) a 5 b 000 c 000 d 56

2 Find the domain of f( ) a (,0) (0, ) c (, ) (, ) (,) (, ) b (, ) (,) (, ) d (, ) (,) (, ) Which describes all real values of for which f ( ) 9 is a real number? a b c 0 d 0 If the minimum value of f( ) occurs at, at which value of does the minimum value of f( ) occur? a b 5 c d 6 5 Which describes the set of all real values of y for which y f ( ) if a y y and y b y y c y y or y f( )? d all real numbers 6 Find the range of f ( ) 9 a, b, c 0, d [0,] If f ( ) k and f ( ) 6, find f () a 68 b c 6 d 56 8 If y and z 5, find an epression for z in terms of y a z = y + b z = y c z = y 9 d z = y The demand d for items priced at p dollars per item is given by d( p), and the revenue R of p selling d items at price p is given by R pd Find R as a function of p 9000 p 899 a Rp ( ) b Rp ( ) p 899 p 9000 p c Rp ( ) d R( p) p p p p p 0 Find the verte of the parabola a, 5 b (, 8) f ( ) 6 5 c (, ) d, 8 Find the solutions to the equation 8 a 0 b 0 8 c 8 0 d 0 Which equation has 0,, as its solution set? a 0 b ( )( ) 0 c ( )( ) 0 d () 0

3 What are the zeros of the function f ( ) 9 8? a =, =, = c =, =, = b = 0, =, =, =, = d = 0, = 9, =, = Which cubic function has 0,, and 5as zeros? a f ( ) 0 b f ( ) 0 c f ( ) 0 d f ( ) 0 5 If the domain of f ( ) 9 9 is all real numbers, what is the solution set for f( ) 0? a9, 9 b, c,, 9 d,, i, i 6 Factor f ( ) 8 completely over the comple numbers a f ( ) ( )( )( )( ) c f ( ) ( )( )( i)( i) b f ( ) ( )( )( 9) d f ( ) ( i)( i)( 9) Which quadratic function has a corresponding graph with a verte at (, ) and contains the point (,5)? a f ( ) ( ) b f ( ) c f ( ) ( ) d f ( ) 8 Simplify a ( a ) 8 a b 9 Simplify a b a b a 5 6a b b 5 a c 5a b c a d 8 6a b d a 6a b 0 Simplify a y y y 5 6 b 8y c y 8 5 d y 5 Simplify, assuming that the variables represent positive numbers: a a a b 8a a 6 a c a a d Not possible If a 5 5 then? b 6 c d 5 5 a a5 If for all real values of such that 0, find the value(s) of a 5a a a 5 b a 0 c a 5 d a5, a

4 Rewrite using logarithmic notation: M y a log y M b log y M c log M y d log M y 5 If log a, then? b c d 6 6 Epress as a single logarithm: log0 log0 y 6log0 z 8log0 t 9y a 6log 0( yzt) b log 0(6 y 6z 8 t) c log0 68 zt If , find the value of a b 0 c d 0 8log ( y z t) d 8 If ln( ) 0, find the value of a 5 b c e d 0t e 9 The function T( t) 0 86 gives the temperature of an object in a room t hours after midnight Find t when T = 85 Give your answer rounded to the nearest tenth a b 0 c 00 d 6 0 Simplify: i i i i a i b 5 i c 5 i d i The product of two comple numbers is i One of the comple numbers is i Find the other comple number a 5i b 0 i c 0 i d i If z is a comple number with z its comple conjugate, and z z 6i, find z a 6 b i c i d i If the nth term of a sequence is given by ( ) n n, find the ( n )th term n a ( ) n n b n n n c ( ) n n n n d ( ) n n n A certain arithmetic sequence has a 56 and a 6 Find a a b 8 c d 8

5 5 If,, are the first three terms of a geometric sequence of positive real numbers, find the common ratio r a b c 5 d 6 Which of the following could be the seventh term in a geometric sequence with a and a 9? a b 8 c d 9 A recursive sequence is given by a i, and an i an for n, where i Find a 5 a i b i c i d i 8 Evaluate (6n ) n a 58 b 60 c 58 d 5 9 A stack of cans has 60 cans in the bottom row, and each row above the bottom row has two less cans than the row below it Find the total number of cans in the first 0 rows, starting with the bottom row a 600 b 580 c 560 d Evaluate 9!! 6! a b 8 c d If A and B 0, find the entry in the first row and the first column for A 5B a b 5 c d k If , then k =? a b c 8 d 5 If a b ad bc c d and k =0, then k =? a 5 b c 0 d 5y 5 Give the z-value of the solution to the following system: z yz a b c d 5

6 55 Which of the following is an augmented matri for the given system? 5 5 a b c 5 5y y d 5 56 Which of the following gives the solution to? a b [,] c [,] d (, ] [, ) 5 Which of the following graphs are functions? I II III IV a I only b I and II only c I, III, and IV only d all of I, II, III and IV 58 For the vectors a, and b, 5, find ab a 5,8 b c d 59 For the vectors a, and b, 5, find ab a 5,8 b c d 60 If the point (, ) is reflected across the -ais, then translated by the vector, coordinates of the resulting point? a (5, 5) b (,) c (, 5) d (5,) 6 Given sets A,, and,8, 0 B, which of the following is a function from A to B? a f ( ) b f ( ) 6 c, what will be the f ( ) d f ( ) 6 Data is collected on two variables, and y, shown in the table below Which of the following equations describes the relationship between and y? y a y y 0log c y 0 y log 6 b 0 d 0 0 6

7 Answers to Advanced Algebra Topics COMPASS Review c b d c 5 a 6 a d 8 a 9 b 0 b d d a b 5 c 6 b b 8 b 9 c 0 d d b c d 5 b 6 c b 8 b 9 d 0 c c c d d 5 c 6 c a 8 a 9 d 0 b b c c d 5 d 6 c b 8 a 9 d 50 b 5 c 5 a 5 d 5 b 55 b 56 c 5 b 58 a 59 c 60 d 6 d 6 b Solutions to Advanced Algebra Topics COMPASS Review 6 f () () () 6 5, and g( ) ( )( ) ( )( 6), so f() g( ) 5 f ( ) g( ) h( ) ( ) ( )( 5) ( ) ( 0 5) g( ) 8 9 6, then use long division to simplify: f ( ) Quotient is ( g f )( ) = g( f ( )) = g( ) = 9 ( ) ( ) = ( 8 ) 9 6 (9 6) 0 (9 ) 8 = 5 6 f ( ) ( ) ( ) ( 6 9) ( ) 6 6 f ( h) f ( ) ( h)(( h) ) ( ( )) ( h)( h ) ( ) h h h h 6h h h

8 g() () 8, so f[ g ()] = f ( ) = ( ) = 9 = 8 y Reverse the roles of the and y to form the inverse relation y This is the inverse relation Now solve for y to epress as a function y y y ( ) f 9 g ( ) is the inverse function for f( ), so follow the same procedure as in question 8 y6 Now reverse the roles of the and y to form the inverse relation 6y This is the inverse relation Now solve for y to epress as a function y can also be written as y, so g( ) f ( ) The and y values swap, so an ordered pair in f ( ) is (,) The input and output values swap, so V (50) will be the value of the input that will have an output of 50 for the function Vt () t t 56 months 5 The domain is all real values of, ecept those that would make the denominator equal zero 0 ( )( ) 0,, so the domain is (, ) (,) (, ) 9 must be greater than or equal to zero in order to get real-valued outputs 9 0, so the domain is [, ), or using set notation, The graph of f( ) will be the graph of f( ), but shifted right units and down unit Therefore, the -coordinate of the minimum will be shifted right units, so it will occur at 5 5 One way to answer this is to eamine the graph to determine the range of f( ) The graph has a horizontal asymptote at y, and when or, the graph is above the horizontal asymptote, so y When is between and, the highest output occurs at 0, with y f(0) Therefore the range, or set of real values of y, is y y or y 6 The outputs of the radical epression 9 will be 0 or larger, and then the term will decrease these values by Therefore the range is y, or in interval notation,, Use f ( ) 6 to find k : Then 6 k( ) ( ) 6 8k k f () () () Substitute y in place of in the equation z 5, then simplify: z ( y) 5 zy 8

9 9 Substitute 9000 in place of d in the equation R pd, then simplify: p ( ) p p p p p p p p R p p p p p p p p p 0 Method : Complete the square to obtain the form f ( ) 6 5 The verte is (, 8) ( ) 5 f ( ) a( h) k, where the verte is ( hk, ) : ( ) 5() ( ) 8 b Method : Use = to find the -coordinate, then substitute into f( ) to find the y-coordinate: a 6, and y f( ) ( ) 6( ) 5 = 8 The verte is (, 8) () Since the equation is quadratic, write it in standard form and use the quadratic formula: 0 8, and simplifies to b b ac a 0 or 0 ( 8) ( 8) ()() () Many equations could have this solution set, but the choices given are all polynomials in factored form, set equal to zero This allows us to use the fact that if a is a zero of a polynomial function, then ( a) is a factor of the polynomial This allows us to write the equation ( )( ) 0 This function is factorable by grouping: f ( ) ( ) 9( ) ( 9)( ) ( )( )( ) Set f( ) = 0 to get zeros of,, and The zeros will create factors of ( )( )( 5) Multiply these out to get f ( ) 0 5 Set and solve to get 9 0 or 9 0 The solutions to 9 0 are nonreal, and since the domain is specified as real numbers, we discard the nonreal solutions This means we have only 9 0 ( )( ) 0 6 Factoring as a difference of squares we get f ( ) 9 9 The zeroes of 9 are, as in question 5, and the zeroes of 9 satisfy i Using the fact that if a is a zero of a polynomial function, then ( a) is a factor of the polynomial, we have the factored polynomial f ( ) ( )( )( i)( i) 9

10 Use the form of a parabola: f ( ) a( h) k Insert the coordinates of the verte for h and k: Now use the other ordered pair to find a: Now create the function using the a value: 8 Multiply the eponents together to get a f ( ) a( ) ( ) 5 a( ) ( ) a f ( ) ( ) ( ) or f ( ) 9 Multiply the coefficients, and add the eponents on the like variables: 6a 5 b = 6a b 0 Simplify the outside power first, then the negative eponents, and then combine the like variables 6 5 y 9 8 y = 5 8 y y 6 9 = y 8 5 Write the radicals as fractional eponents, then use properties of eponents to simplify a a a a a a a a a a a a Start by writing both sides of the equation as powers of 5, then solve for () Start by simplifying both sides as powers of, then solve for a a 5a a5 a 5a a 5 a 6a 5 0 ( a5)( a) 0 a5, a Use the definition a y log y to get log M y a 5 log a, 6 Use the properties log( ab) log( a) log( b) log log y 6log z 8log t = log log( a) log( b), and log alog b 0 0 y 6 0 z 8 0 t log log log log = 6 8 y z t = 6 8 log 9y log z t log 9 log log log = log a 9y zt ln( ) 0 9 Solve for t: 0t e 0 e 5 5 0t e t ln 6 hours 0t e t e 86 5 ln 0t 86

11 0 Remember i so i Then i i i i = i i i i i = i 6i 8i i i i i 6i 8i i i = i 6i 8( ) i ( ) = i 6i 8 i = 5 i = Write ( i)( a bi) i, then find a bi i i i a bi i i i = i i 6i = 0 i = 0 i Let z a bi, so z a bi, and we have ( a bi) ( a bi) 6i Simplify the left side to get a bi 6i a 0 and b, so z i Substitute n + in place of n in the epression a n ( n) ( n ) n n ( ) ( ) ( n ) n n ( ) n n to get n You can use the formula for the nth term in an arithmetic sequence: an a ( n )( d) 6 56 ( )( d) d d Since we now have a and d, use the formula to find a : a 56 (6)( ) 8 n 5 Use the formula for the nth term in a geometric sequence: an a r, with a and a : r r r But since we are told that all the terms in the sequence are positive, this rules out the possibility of a negative value for r, so r n 6 Use the formula for the nth term in a geometric sequence: an a r to get a system of two equations: a r ar, and 9 a r 9 ar We can substitute in place of ar in the second equation to get 9 r r Since all of the choices given are positive values for the seventh term of the sequence, r must be positive, so we use Now we can use r to find a, then use the formula for the nth term to find a : 5 a a, and a

12 a i 8 a i a i( i) i i i a i a i( i) i i i a i a i( i) i i i a i a i( i) i i i 5 n (6n ) (6) (6 ) (6) (6 ) = We can see that this is a sum of the first terms in an arithmetic sequence with common difference d 6, so use the formula for the sum of the first n terms in an arithmetic sequence: S a a 5( 5) 58 9 We want to find the sum of the first 0 terms in an arithmetic sequence, with first term a 60 and common difference d Use the formula for the nth term in an arithmetic sequence to find the number of cans in the 0 th row: a0 60 (0 )( ) cans Then use the formula for the sum of the first n terms in an 0 arithmetic sequence: S a a 5(60 ) 50 cans ! 986! 98! 6! 6! 8 5 Multiply each entry in B by 5, then add the corresponding entries in A and 5B The entry in the first row first column is 8 5 k Multiply the two matrices on the left side of the equation to get 5k 5 The entries in the two matrices must all be the same, so we have k 6 and 5k 5, which imply k 5 Apply the definition a b ad bc c d to the left side of the equation to get k solve for k to get k 0 k 0, then 5y 5 One method is to rewrite the system as: z y z Then multiply the nd equation by and add the result to the st equation to eliminate, which gives 5y z Using this result with the original rd equation, we have a system of two equations with two 5y z variables: y z We want eliminate y now, so multiply the top equation by and the bottom equation by 5 When we add them together and solve for z, we get z =

13 55 First, write the terms in the same order to get 5 y y Then, write a matri where each row gives the coefficients of an equation One type of notation for an augmented matri uses a dashed line to indicate the augmentation of the coefficient matri with the matri 5 of constant terms: 56 One way to proceed is multiply both sides of the inequality by, but this requires us to switch the direction of the inequality symbol: Then one method to finish the solution is to write the inequality with zero on one side and eamine the sign changes of the other side: 0 ( )( ) 0 The product on the left side will be equal to 0 when either factor is 0, which occurs when or when The product on the left side will be less than 0 when one of the factors is negative and the other is positive Since for all values of, the factor is larger than the factor, the positive factor with be and the negative factor will be This means 0 and 0, which implies and, which means So ( )( ) 0 is true when, or when is in the interval, 5 The only graphs that pass the vertical line test are I and II 58 Distribute the through the components of b and then add the corresponding components of a and b : ab,, 5, 6,5 5,8 59 ab,, 5 ()() ()( 5) 5 60 When the point (, ) is reflected across the -ais, we get the point (,) Then, if we translate the point (, ) by the vector,, we move right and down to end up at (5,) 6 A function from A to B must map each element of A to one element of B (but it is not necessary to use all of the elements of B) There are many possible functions from A to B, so the only way to answer this question is to find which of the choices works We can try evaluating each function using the elements of A to determine whether or not we get values that are in B a f () () 0, which is in B f () (), which is NOT in B, so this choice is incorrect b f () 6() 8, which is in B f () 6() 0, which is in B f () 6(), which is NOT in B, so this choice is incorrect c f (), which is in B f () 8, which is in B f (), which is NOT in B, so this choice is incorrect Choice d had better work d f () (), which is in B f () () 8, which is in B f () () 0, which is in B, so this choice is correct

14 6 As in question 6, there are many possible equations that could relate the variables We can try the equations to see which one works for each pair of data values a y works when 0, but not for 0 since So this choice is incorrect b y 0log 0 When when when works for each pair of data values: y 0log 0log 0() 6, and y 0log 0log 0() 6, and y 0log 0log 8 0() 6 So this choice is correct 0 (Choices c and d do not work for any of the pairs of data values)