Mesa College Math 6 - SAMPLES Directions: NO CALCULATOR. Write neatly, show your work and steps. Label your work so it s easy to ollow. Answers without appropriate work will receive NO credit. For inal answers, be sure to simpliy all radicals and ractions. #. Find the domain o each unction. Epress your answer in interval notation. 4 a) ( ) 4 5 b) g( ) 4 5 c) h( ) 0 8 #. Find ( a + h) ( a). Completely simpliy your result, where: h a) ( ) + + 5 b) ( ) 5 c) ( ) #. Find ( g)( ), and speciy the domain o g using interval notation, where: a) ( ) + and g( ) b) ( ) and g( ) 4 + c) ( ) g( ) + #4. Find all such that ( ) 0. Write your answer using interval notation, where: 4a) 4 ( ) + +. 4b) 4 ( ) 8 + 4 4c) ( ) 9 #5. Find the equations o all vertical and horizontal asymptotes or : 5a) ( ) + + 5b) g( ) 4 5c) h( ) 5 + 4 9 #6. Epress as a single logarithm. Assume all values are properly deined #7. Solve: 6a) b log + 4log y 6log y 6b) log log b y + b b + 7b) ( ) log log log (4 ). 7a) 7d) log log ( 6) log log ( + ) 7c) 5 7 + + + 7e) 7 9( ) log 5 7) ( ) log 5 7g) 0.4 e 7h) log 5
#8. Suppose A 4, B 5, and C 0. Find the ollowing: 8a) AB 8b) A - C 8c) CB 8d) BC 8e) A - #9. Use augmented matrices and Gaussian elimination to solve: 9a) + y z + y z y + z 0 9b) y + z 9 + y 4 5y + 5z 7 9c) + y 5y z 4 4 + z 7 #0. Maimize P 0 + 7 y, given the constraints 0 60, 0 y 45, and 5 + 6y 40. Sketch the graph o the constraint inequalities. Label the critical vertices on the graph. #. An accountant prepares ta returns or individuals and or small businesses. On average, each individual return requires hours o her time, and hour o computer time. Each business requires 4 hours o her time and hours o computer time. Because o other business considerations, her time is limited to 40 hours, and the computer time is limited to 00 hours. I she earns a proit o $80 per each individual return, and a proit o $50 on each business return, how many returns o each type should she prepare to maimize her proit? #. SOLVE each over the comple number system. a) c) 5 6 b) 4 7 + d) 4 5 + 4 0 + 7 + + 5 0 #. Sketch the graph o each polynomial unction labeling the zeros and the y-intercepts or each. a) y 5 4 + b) 4 y + #4. Let F() be a polynomial unction with rational coeicients. Solve each, over the comple numbers, i necessary, using the given hints. 4a) + 60 0, where 5 is a root. 4b) 4 7 8 6 0 +, where is a root #5a) Write a third degree polynomial equation with rational coeicients, that has 5 and as its roots #5b) Write a ourth degree polynomial equation with integer coeicients that has i (where i ) and - as roots, where - is a double root. (also called: - has a multiplicity o two.)
#6. The below graph represents part o a unction y (). Sketch each: 6a) y (+) 6b) y () 6c) y ( ) 6d) y ( ) 6e) the relection o () over the y-ais 6) the relection o () over the -ais #7. Suppose () is a continuous, one-to-one unction, ind: 7a) ( ), i () 5, and () is an even unction 7b) ( ), i () 5, and () is an odd unction #8. Sketch the graph o each piece-wise unction. Label the coordinates o each verte and/or each point o discontinuity. 8a), i < y, i 8b), i < y +, i, i > #9. Resolve each into its partial ractions. + 9a) 7 5 9b) + 7 + ( )( + ) #0. A new drug is injected into a patient. The number o milligrams remaining in the patient s bloodstream t hours is modeled by: D( t) 60 0.t e. (a) How many milligrams were initially injected? (b) How many milligrams o the drug remain ater 5 hours? (c) When will there be one-orth o the initial dosage in the patient s bloodstream? #. Students in a math class took a inal eam. As part o a study, they were tested each month thereater to determine how much they remembered. The ollowing ormula was derived: S( t) 77.4 0 log( t + ), t 0. Where S(t) represents the score (as a percent), and t represents months. (a) What was the average initial score on the eam? (b) What was the average score ater 9 months?
ANSWERS to Math 6 Challenge Eam SAMPLES Disclaimer: A ew o these answers may (will) be wrong! There s a myriad o reasons why: I m stooped, my ingers are at, I m dysleic, I m tired, the summer sun has baked my brain. Please, email me your suggested solution(s), I ll double check it, and get back to you. Thanks. Larry Foster: laoster@sdccd.edu NOTE: I have NOT yet included the GRAPHS someday. *********************************************************************************************** a) Domain ) ( ) b) Domain ( 5, ) ( 5, 8 c) Domain ( ) 4,, a) -a + h b) 4a 5 + h c) a + ah 5, a) b) g +, domain (, ) (,0) (0, ) g +, domain ( (, ), ) (, ) c) g +, domain ( (, ), ) (, ) 4a) ( ) > 0, when [, ] 4b) ( ) > 0, when [, ] [, ) 4c) ( ) > 0, when [, ] (, ) 5a) ( ),, vertical Asymptote: none, horizontal Asymptote: y 0, + Point o Discontinuity (aka: hole ): (0, ) 5b) 5c) + g( ),, vert: 4, horiz: 4 + y, hole @ (, ) 5 + h( ), 0, vert: ±, horiz: 0 ( + )( ) y, hole @ ( 0, ) 6a) log( y) 6b) 7a) { }, 0 rejects 7b) { }, 7 6 rejects 7c) { 5 } 7d) { }, -9 rejects 7e) { 8} 7) { } 7g) ln 0.04 this is best possible w/o calculator 7h) { 5 } 8a) 9 9 8b) 0 6 8c) 0 7 8d) not deined 8e) 4
{ } { } { } 9) ater a LOT o work: 9a) (,, ) 9b) (,, ) 9c) ( 4,,) Remember: the parenthesis inside o the solution set is mandatory! 0) See below or sketch. corners: (0,0), (0,45), (60,0), 0,45), (60,0) The ma value o P occurs when 60 and y 0. ) Let # o individual returns, y # o business returns Objective unction: P 80 + 50y Constraints: 0 y 0 + 4y 40 + y 00 corners: (0,0), (0, 50), (40,0), 80,0) Ma proit occurs @ (40,0 ) she should prepare 40 individual returns, and 0 business returns. a) { ±, ± i} b) {,} c) { 8, } 8 d) { } i,, ± # See below or sketches. Using a combination o: Rational Root theorem, synthetic division, actoring, and quadratic ormula: a) y ( + )( )( ) zeros:,,, y-intercept: ( ) 0, + check out it s etreme behavior b) y ( ) ( + ), zeros: 0, -, (multiplicity o ), y-int: (0,0) #4 see hints rom # 4a) ( 5)( + )( + 4) 0 { 5,,4} 4b) ( )( )( + + ) 0 {,, ± i} 5a) remember with rational coeicients, each irrational and comple root must include its conjugate. ( 5)( + 5)( ) 0 0 + 0 0 5b) 4 ( i)( + i)( + )( + ) 0 + + 0 + 8 + 0 #6 graphs later. Each corner o the shape should be modiied as indicated below. Then, connect the dots. 6a) shit entire graph let units. (subtract rom -values, keep y-values the same) 6b) (divide -values by, keep y-values the same) 6c) swap all y that is: (-,4) (4,-) (0,) (,0) (,) (,)
(5,0) (0,5) 6d) shit entire graph down units: (keep -value the same, subtract rom y-values) 6e) (change all -values to their opposite, keep y-value the same) 6) (keep -value same, change y-values to their opposite) 7a) ( ) 5 7b) ( ) 5 #8 see below or sketches (someday) 9a) 7 + ( + ) ( 5) 5 0 9b) + + ( + ) + o course you don t need this last raction 0a) 60 mg 0b) 60e - mg, or 60 e mg 0c) ln( 4) hours, or 0. ln(4) 0. hours. a) The initial average score was 77.4% b) Ater 9 months, the resulting average score was 57.4%