MAC 1105 Review for Exam 4. Name
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1 MAC 1105 Review for Eam Name For the given functions f and g, find the requested composite function. 1) f() = +, g() = 8-7; Find (f g)(). 1) Find the domain of the composite function f g. 9 ) f() = + 9; g() = + 6 ) For the given functions f and g, find the requested composite function value. ) f() = - 6, g() = + 9; Find (g f)(-). ) The function f is one-to-one. Find its inverse. ) Determine the equation for the inverse function of = ( + ) - 8. ) 5) f() = + 5 5) 1
2 Find the inverse function of f. State the domain and the range of f and of f -1. 6) f() = ) f-1() = Domain of f: Range of f: Domain of f-1(): Range of f-1(): The graph of a one-to-one function f is given. Draw the graph of the inverse function f-1 as a dashed line or curve. 7) f() = + Find the equation and the graph of the inverse and its domain and range. 7) f-1() = Domain: Range: Use the graph of the given one-to-one function to sketch the graph of the inverse function. For convenience, the graph of = is also given. 8) 8) (-, 1) 5 1 (1, ) (0, ) (-, -)
3 Solve the problem. 9) The function D(h) = 8e-0.h can be used to determine the milligrams D of a certain drug in a patient's bloodstream h hours after the drug has been given. How man milligrams (to two decimals) will be present after 11 hours? 9) Determine the eponential function whose graph is given. 10) 0 10) A) f() = - B) f() = C)f() = -- D) f() = - a) Graph the function using ke points; b) Find the domain and the range of the function. 11) f() = 1 11) Solve the equation without using a calculator. Show all the necessar steps. 1 1) 5 = 65 1)
4 1) (7 - ) = 1 1) 1) ( - )= 6 1) Convert to logarithmic form. 15) 7 = 15) 16) e = 9 16) Convert to eponential form. 17) log 1/ 7 = - 17)
5 Change logarithmic epression to eponential epression. 18) logb 9 = 18) Find the value of the epression without using a calculator. 19) log ) Use a calculator to find the natural logarithm correct to four decimal places. 0) ln 0) Use the properties of logarithms to find the eact value of the epression. Do not use a calculator. 1) log log 5 1) Using the properties of logarithms, evaluate the epression. ) ln e. ) ) Given logb = 0. and logb = 0.8 use the properties of logarithms to find logb 7 16 without using a calculator. ) 5
6 Write as the sum and/or difference of logs. Do not use eponents. ) log 6 1 m n ) 5) log 19 9 nm 5) Write the logarithmic epression as a sum and difference of logarithms. Write the eponents as factors. 6) loga + 5 ( -) 6) Epress as a single logarithm. 7) (log a - log a ) + log a z 7) Write epressions as a single logarithm. 8) ln 16 - ln( - - ) 8) 6
7 Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round our answer to two decimal places. 9) Evaluate log 5. 9) Graph the function using a graphing utilit and the Change-of-Base Formula. 0) = log6 1 0) Find the domain of the function. 1) f() = log ( - ) 1) ) Find the inverse of the following functions: a) f() = + 7 b) g() = ln( + ) ) Solve the equation. ) log = 5 ) 7
8 ) = log/ (Do not use a calculator. Convert to eponential form first) ) Solve the equation. 5) log ( + ) - log ( - ) = log 5) Solve the given logarithmic equation. 6) log( - ) - log( - 5) = 6) Solve the equation. 7) log + log( - ) = 7) 8) log = log 5 + log ( - ) 8) 8
9 Solve the equation. If necessar, round our answer to two decimal places. 9) ( - 1) = 1 9) Solve the given eponential equation. 0) 5 t - 1 = 75 0) Solve the eponential equation. Epress the solution set in terms of natural logarithms. 1) + = 5+5 1) Find the time needed to double our mone. Epress our answer rounded to decimal places. ) How man ears will it take for $10,000 to double in value if it earns 7% compounded continuoul? Round answer to two decimal places. ) Solve the problem. ) The size P of a small herbivore population at time t (in ears) obes the function P(t) = 500e0.t if the have enough food and the predator population stas constant. After how man ears will the population reach 500? ) 9
10 ) Strontium 90 decas at a constant rate of.% per ear. Therefore, the equation for the amount P of strontium 90 after t ears is P = P0 e-0.0t. How long will it take for 15 grams of strontium to deca to 5 grams? Round answer to decimal places. ) 5) A certain radioactive isotope decas at a rate of 0.% annuall. Determine the half-life of this isotope, to the nearest ear. 5) 6) A certain radioactive isotope has a half-life of 555 ears. Determine the annual deca rate, k. 6) Answer the question. 7) Define the number e. A) The number approimatel equal to.7. lim B) The number defined b e = n n n in Calculus. 7) C)The number that the epression, n n, approaches as n. D) All of the above. 10
11 Answer Ke Testname: 105REV_E_011 1) - 1 ) { -6} ) 5 ) = + 8-5) f-1() = - 5 6) f -1() = 5 + ; domain of f: { -5}; range of f: { } - 7) f-1() = - for 0 ; Domain: 0 ; Range: ) ) 0.1 mg 10) B (-, -) (, 0) (1, -) (, 1) 11
12 Answer Ke Testname: 105REV_E_011 11) Ke points: (-1, ), (0, 1), (1, 1 ); b ) Domain: (-, ); Range: (0, ) ) {-} 1) {} 1) = or = - 15) log 7 = 16) ln 9 = 17) ( 1 )- = 7 18) b/= 9 19) - 0) ) 1 ) 8. ) logb 7 16 = 0. ) log log 6 m - log 6 n 5) 1 log log 19 n - log 19 m 6) loga + 1 log a( + 5) - loga( - ) 7) log a z 8) ln( 8 5 ) 9).9 1
13 Answer Ke Testname: 105REV_E_011 0) Use a graphing calculator to graph = log, or prepare a table of values with ke points: (1/6, -1),(1, 0) and (6, 1) log ) { > } ) a) f-1() = log( - 7) b) g-1() = e - ) {} ) ( ) = = ( )-5 ---> = -5 5) {7} 6) = 6 7) {7} 8) {0} 9) {0.9} 0) t = 1) 5 ln 5 - ln ln - ln 5 ) 9.90 ears ) 7. rs ) 5.0 ears. 5) 7 r 6) 0.15% 7) D 1
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