Question

The graph shows the function f(x) = 3* What is the value of f-1(x) at x = 3?

Answer 1

1

I've attached a screenshot from my graphing calculator of
`f(x)` in blue and
`f^(-1)(x)` in red. Notice how the red line (inverse function) has (3, 1)

Explanation:

`f(x) = 3^x\\f^(-1)(x) =\ ?\\`

We must first figure out the inverse function of
`f(x)` which is
`f^(-1)(x)`

`y = f(x) = 3^x\\y = 3^x\\log\ y = log\ 3^x\\log\ y = x * log\ 3\\x = (log\ y)/(log\ 3)\\`

We say y = f(x) to begin with, but after find x = ...

we must 'swap' x and y

`x = (log\ y)/(log\ 3)\\y = (log\ x)/(log\ 3)\\`

`f^(-1)(x) = (log\ x)/(log\ 3)`

`f^(-1)(3) = (log\ 3)/(log\ 3) = 1`