For the function f(x)=2+3lnx/4-lnx a)find f^-1(x) b)check that f^-1(f(x))=x

Question

For the function f(x)=2+3lnx/4-lnx

a)find f^-1(x)

b)check that f^-1(f(x))=x

Answer 1

I think the function is

f(x) = ( 2 + 3 lnx ) / ( 4 − lnx ) because the result looks much more beautiful and elegant.

Then

y = ( 2 + 3 lnx ) / ( 4 − lnx )

Hence

x = ( 2 + 3 lny ) / ( 4 − lny ) ⇒ 4x −x ln y = 2 + 3 lny

ln y = (4x−2) / ( x +3 ) ⇒ y = e^((4x−2) / ( x +3 ))

There f^-1(x) = e^((4x−2) / ( x +3 ))

Because this is what Math is : Elegance and beauty

Answer 2

Start with y=f(x). y=2+3ln(x)/4- ln(x) Solve for x.

Using properties of log 3lnx/4 = ln x^3/4 and ln x^3/4- ln x = ln(x^3/4/x) = ln ( x^-1/4)

So y=2+ln(x^-1/4) Now we raise both sides to the e power giving

e^y= e²*e^{ln(1/x^4)} = e²*1/x⁴= e²/x⁴

e^y=e²/x⁴ Solving for x⁴ we have x⁴= e²/e^y = e^(2-y) and taking the fourth root

x= (e^(2-y))^(1/4)

Switch x and y, and you have the inverse function f^-1(x) = ( e^(2-x))^(1/4) or in words the inverse function is the 4th root of e raised to the 2-x power.

1. Replace f(x) with y
2. Solve for x
3. Switch x and y
4. Now, the process of solving has transformed f(x) into f^-1(x)
5. Replace y with f^-1(x)

To check it , you simply put in f(x) for x in the original equation:

f^-1(2+3ln(x)/4-ln(x)) = (e^{(2-2-3ln x/4 +ln x)})^(1/4)