Question

f(x)=3ln(x) Find inverse function & the domain of inverse.

Answer 1

to find inverse: let f(x) = y and invert x and y.

so y = 3ln(x) becomes x = 3ln(y).

Now solve for y to put it back in the form of a f(x)

ln(y) = x/3

inverse operation of ln(y) is e^y

e^(ln(y)) = e^(x/3)

y = e^(x/3) = g(x) = inverse of f(x)

Answer 2

oh okay thank you!

Answer 3

inverse operation of ln(y) is e^y

e^(ln(y)) = e^(x/3)

Answer 4

I ment this step

Answer 5

it is just the definition of the logarithmic function.

For:
\[y = \log_b(x)\]
\[x = b^y\]

\[\ln(y) = \log_e(y)\]

Answer 6

Wait but right now my paper says x/3=e^y right

Answer 7

So would it be log e x/3=y

Answer 8

that is the truth. those two equations are equal

Answer 9

oh. i think i see the misunderstanding:
the point of that step was that:
\[\ln(e^y) = y\]

Answer 10

which is what i meant by inverse operation

same way that x = sin^(-1)(y) is the inverse of y = sin(x), used to find the angle x

Answer 11

and that sqrt is inverse of square

Answer 12

So would it be

Answer 13

log e x/2 = y?

Answer 14

yup. assuming that 2 is a typo :)