Question

. Find the inverse of each function.

a. f(x) = 2x−3
b. g(x) = 2 log(x − 1)
c. h(x) = ln(x) − ln(x − 1)
d. k(x) = 5 − 3^−x/2

Answer 1

a.
`f^(-1)(x)=(x+3)/(2)`

b.
`g^(-1)(x) =e^{(x)/(2)}+1`

c.
`h^(-1)(x)=(e^(y))/(e^(y)-1)`

d.
`k^(-1)(x)=-(2log(5-y))/(log(3))`

Explanation:

Here is the procediment for each case.

a.
`f(x)=2x-3` as we know
`f(x)=y` then
`y=2x-3` finding the expresion of x, we find the inverse of the function, then:

`y=2x-3\\y+3=2x\\\\(y+3)/(2)=x`

Then:

`f^(-1)(x)=(x+3)/(2)`

b.
`g(x)=2log(x-1)` for this we have to remember that the inverse function of the log is the exp, then:

`g(x) = 2 log(x-1)\\y=2log(x-1)\\\\\\(y)/(2)=log(x-1)\\\\e^{(y)/(2)}=e^(log(x-1))\\\\e^{(y)/(2)}=x-1\\e^{(y)/(2)}+1=x`

Then the inverse function is:

`g^(-1)(x) =e^{(x)/(2)}+1`

c. In this case we have to also remember the relation between the e and the ln, then:

`h(x) = ln(x)-ln(x-1)` with the properties of the ln we have:

`h(x) = ln(x)-ln(x-1)=ln\left((x)/(x-1)\right)` now finding the inverse function we have:

`h(x)=ln\left((x)/(x-1)\right)\\\\y=ln\left((x)/(x-1)\right)\\\\e^(y)=e^{ln\left((x)/(x-1)\right)}\\\\\\e^(y)=(x)/(x-1)}\\\\e^(y)(x-1)=x\\e^(y)x-e^(y)=x\\e^(y)x-x=e^(y)\\x(e^(y)-1)=e^(y)\\x=(e^(y))/(e^(y)-1)`

then:

`h^(-1)(x)=(e^(y))/(e^(y)-1)`

d. in the last one we have:
`k(x) =5-3^(-x/2)` then:

`k(x) =5-3^(-x/2)\\y =5-3^(-x/2)\\y-5=-3^(-x/2)\\5-y=3^(-x/2)\\log(5-y)=log(3^(-x/2))\\log(5-y)=-(x)/(2)log(3)\\-2log(5-y)=xlog(3)\\-(2log(5-y))/(log(3))=x`

then:

`k^(-1)(x)=-(2log(5-y))/(log(3))`