Unit 7 Review - Radicals and Rational Expressions

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1 Name: Class: Date: D: A Unit 7 Review - Radicals and Rational Epressions. Simplify 8 4. b. 8 6 True or False:.. For any integer a:;:. 0, a n = an'. For any integer n > 0 and any positive real number a, alln =~. 4. The volume ofa dodecahedron (a solid with regular pentagons as faces) is V ~ 7.66a, where a is the length ofan edge. Find the edge length ofa dodecahedron whose volume is 000 cubic centimeters. 5. Find values for a, b, and c such that T =«(7) a )by. Then simplify the epression T/. Simplify the epression. 6. V 8 + V 04 Simplify the epression. Assume au variables are positive. 8. V6 y 7 V The area ofa circular park is 0 4 square feet. The radius r (in feet) can be approimated by the model r =( ~r where A is the area ofthe park (in square feet). Approimate the radius ofthe park. Use.4 for n. Round your answer to the nearest whole number. 0. Let r() = and s() -. Find s(r(».. A discount function D() that takes 0% off an entire purchase can be given by D() == 0.9O where is the amount ofthe entire purchase. A ta function T() that adds a ta of 0% to an entire purchase can be given by T() =. lo. Eplain whatd(t() and T(D(») represent. Then compare the two compositions.

2 Name: D: A Let f() = ( ) andg() =_. Perform tbe indicated operation and state tbe domain.. ftg(». g(f(» 4. g(g(» 5. Find the inverse ofthe relation (-5, 8), (8, -5), (-6, ). 6. Write an equation for the inverse ofthe rel~tion y = -ll Sketch the graph ofthe function f() =_ -. s the inverse ofj{) a function? Open-ended: 8. Eplain how the inverse ofa function is determined algebraically. Give an eample to eplain the steps you follow. Grapb tbe function. Tben state tbe domain and range. 9. f()=-v-- Solve tbe equation. Cbeck for etraneous solutions. 0. X / 4 = 9. Which gives the solution of V = 6 a~ -5 b. none ofthese 7

3 Name: D: A. Solve j5-4.. Solve"J + =~ +. Check for etraneous solutions. 4. For a mammal, the heart rate r (in beats per minute) and brain weight w (in kilograms) are related to the body mass m (in kilograms) by the functions r =4m- 4 and w =9.m /4 Solve the function r = 4m-/4 for m. b. Solve the function w = 9. m 4 for m Eplain how to use the answers from parts (a) and (b) to epress the heart rate r as a function ofthe brain weight w. Then find the function. 5. t takes a train 8 hours to travel from Capital City to Johnson City when it travels at a speed of65 milh. How long would it take the train to go the same distance when it travels 40 milh? h ~ h 8 b. h h 6. The wattage rating W (in watts) ofan appliance varies jointly with the square ofthe current (in amperes) and the resistance R (in ohms). f the wattage is 6 watts when the current is 0. ampere and the resistance is 50 ohms, find the wattage when the current is 0. ampere and the resistance is 00 ohms. 80 watts 7,000 watts b. 90 watts 7 watts 7. Write an equation for the relationship: y varies directly with the square root of and inversely with t 8. The intensity (in foot-candles) oflight received from a source varies inversely with the square ofthe distance d from the source. fthe light intensity is 5 foot-candles from 6 feet, find the light intensity from 9 feet. Round your answer to the nearest hundredth, if necessary. 9. Let f{) = ~:i. Find the asymptotes ofthe graph off, and tell how the graph is related to a hyperbola with equation ofthe form y =!!:. dentify the vertical asymptote(s) ofthe graph ofthe function f{) = < + +&+8 =-4, = -, = - = -4, =- b. none =-4, =-

4 Name: D: A. f() =- X 4 dentify all vertical and horizontal asymptote(s) of the graph of the function.. f() Use the rational function fi) 5 +&-' What is the horizontal asymptote offi)? Give an eample ofanother rational function with a quadratic polynomial in the numerator that has the same horizontal asymptote as f(). b. Write a rational function with a cubic polynomial in the denominator that has the same horizontal asymptote as f() Compare the graphs ofy = 4 and y = nclude - and y-mtercepts, asymptotes, and end behavior in your analysis. Simplify the rational epression, if possible ! Divide the epressions. Simplify the result b Perform the indicated operations. Simplify the result. 4

5 Name: D: A.. ll + 9. Use the ratlona epresson -7+ Factor the epression. Then simplify. ifpossible... () + b. s t he epresson m part a equa to - 76+l05?E ' p am. - -4l + 05 Describe the error in simplifying the rational epression: = + = A company is designing packaging in the shape ofa right cylinder for a new product. They want the radius of the cylinder to be inches less than its height. Find the volume ofthe cylinder in terms ofthe height.. b. Find the surface area ofthe cylinder in terms ofthe height. Write the ratio ofthe volume ofthe cylinder to the surface area ofthe cylinder in simplest form. The company decides to choose between a height of 8 inches or 0 inches. They want to choose the one which will be the more cost efficient. Which should they choose? Eplain. Perform the indicated operation(s) and simplify S + Y z Y z Simplify the comple fraction b Solve the equation. Cheek for etraneous solutions. a :;:: a b. 4 5

6 Name: D: A 45. e 5 e = -:---:--:----. e+5 e-5 (e+5)(e-5) - b Which equation shows inverse variation between and y? -y= b. y= 6 y - Y -=7 47. fy varies inversely with, and y = 8 when = 6, what is y when =4 6 b The variable z varies jointly with the variables and y. When z = 60, = and y = 4. Which equation relates, y, and z? z y+48 z = 5y 80t b. z = 7-y z y 49. What are the equations for the asymptotes ofthe function y = = -, y =-5 =,y =-5 b. =,y 5 =-,y=5 SO. What are the domain and range ofthe function y =~ + - domain: "* -; range: y > b. domain: "* -; range: y < domain: "* ; range: y > domain: "* ; range: y >- 6

7 j Name: D: A 5. Which graph represents the function y = - + 4? + : Y. i \ ' \i i'-.....:== j j ~... c + --.~ ~,.' ;.,!,. :...-L. j j l L--' l h. i j -'/ t 'L.,)'.... -~---- _.. i V, _..., -. : f :.!! :. L l!. i r--~--...! i y t' - ~ ~.".., /' --~"... -.=.. 'Y ~-'--...LJ ~... r ~ -- -_..... ~ : ~+'f" \... i-" -- /~.' _... ""' ' i... ~'-r-- X - 5. Simplify the epression h. + + (-)(-4) (+4) 5. M' u tip y: h. (-)(-5) (+)(+5)

8 Name: 0: A 54. D' 'd V e. -4+4,.. -+ b. ( -) +l 7( + 4) (X + ) ( ) (X -) Simplify ( + 8)( 8) ( + ) +8 b. (-8)( 8) Subtract: b Add: 5 + ;:\ ( +)( + ) b ,;;;;;.;...; X + lo ( + )( + ) Solve + 5 = + l' -7 b Solve ' - b. 5 no solution 8

9 Name: D: A Solve" +4 = + ' -6or -6 b. 6or- 9