SOLUTION: This question has three parts (a,b,c) a) Find the inverse of the function. b) State the domain of the function and the domain of its inverse. c) Use property of inverse function

Question 6480: This question has three parts (a,b,c)
a) Find the inverse of the function.
b) State the domain of the function and the domain of its inverse.
c) Use property of inverse functions to show your inverse is, in fact the inverse of the function.
f(x)=log(4x)+log((3+x)/x^2)

Answer by khwang(438) (Show Source): You can put this solution on YOUR website!

I suppose the base is 10,if not similar solution.
f(x)=log(4x)+log((3+x)/x^2)
Since numbers inside log must be positive,so
4x > 0 and (3+x)/x^2.
Hence, x > 0 and 3+x > 0 (or x > -3).
Therefore,x > 0 and (0, +oo) is the domain of f.

Let y =log(4x)+log((3+x)/x^2) = log (4x(3+x)/x^2) = log (4(3+x)/x)
= log (4 + 12/x).
By def. of log, 4 + 12/x = 10^y
so 12/x = 10^y - 4, we get x = 12/(10^y - 4).
Since x > 0, 10^y - 4 >0 , so 10^y >4 so, y > log 4.
Exchange x,y , we obtain the inverse function g(x) = 12/(10^x -4)

a.The inverse of f is g(x) = 12/(10^x -4).
b. Domain of the inverse function (log 4, +oo)
c. Check f(g(x)) = log (4 + 12/[12/(10^x -4)])
= log (4 + 10^x -4 ) = log (10^x) = x.
and g(f(x)) = 12/(10^[log (4 + 12/x)] -4)
= 12 / (4 + 12/x - 4)
= x.
This shows g is the inverse function of f.
Kenny

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