document.write( "Question 6480: This question has three parts (a,b,c)
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document.write( "a) Find the inverse of the function.
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document.write( "b) State the domain of the function and the domain of its inverse.
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document.write( "c) Use property of inverse functions to show your inverse is, in fact the inverse of the function.\r
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document.write( "f(x)=log(4x)+log((3+x)/x^2)
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Algebra.Com's Answer #3501 by khwang(438)![]() ![]() ![]() You can put this solution on YOUR website! \n" ); document.write( " I suppose the base is 10,if not similar solution. \n" ); document.write( " f(x)=log(4x)+log((3+x)/x^2) \n" ); document.write( " Since numbers inside log must be positive,so \n" ); document.write( " 4x > 0 and (3+x)/x^2. \n" ); document.write( " Hence, x > 0 and 3+x > 0 (or x > -3). \n" ); document.write( " Therefore,x > 0 and (0, +oo) is the domain of f. \n" ); document.write( " \n" ); document.write( " Let y =log(4x)+log((3+x)/x^2) = log (4x(3+x)/x^2) = log (4(3+x)/x) \n" ); document.write( " = log (4 + 12/x). \n" ); document.write( " By def. of log, 4 + 12/x = 10^y \n" ); document.write( " so 12/x = 10^y - 4, we get x = 12/(10^y - 4). \n" ); document.write( " Since x > 0, 10^y - 4 >0 , so 10^y >4 so, y > log 4. \n" ); document.write( " Exchange x,y , we obtain the inverse function g(x) = 12/(10^x -4) \n" ); document.write( " \n" ); document.write( "a.The inverse of f is g(x) = 12/(10^x -4).\r \n" ); document.write( "\n" ); document.write( "b. Domain of the inverse function (log 4, +oo)\r \n" ); document.write( "\n" ); document.write( "c. Check f(g(x)) = log (4 + 12/[12/(10^x -4)]) \n" ); document.write( " = log (4 + 10^x -4 ) = log (10^x) = x. \n" ); document.write( " and g(f(x)) = 12/(10^[log (4 + 12/x)] -4) \n" ); document.write( " = 12 / (4 + 12/x - 4) \n" ); document.write( " = x. \n" ); document.write( " This shows g is the inverse function of f.\r \n" ); document.write( "\n" ); document.write( " Kenny \n" ); document.write( " \n" ); document.write( " |