Answer:
1
I've attached a screenshot from my graphing calculator of [tex]f(x)[/tex] in blue and [tex]f^{-1}(x)[/tex] in red. Notice how the red line (inverse function) has (3, 1)
Step-by-step explanation:
[tex]f(x) = 3^x\\f^{-1}(x) =\ ?\\[/tex]
We must first figure out the inverse function of [tex]f(x)[/tex] which is [tex]f^{-1}(x)[/tex]
[tex]y = f(x) = 3^x\\y = 3^x\\log\ y = log\ 3^x\\log\ y = x \times log\ 3\\x = \frac{log\ y}{log\ 3}\\[/tex]
We say y = f(x) to begin with, but after find x = ...
we must 'swap' x and y
[tex]x = \frac{log\ y}{log\ 3}\\y = \frac{log\ x}{log\ 3}\\[/tex]
[tex]f^{-1}(x) = \frac{log\ x}{log\ 3}[/tex]
[tex]f^{-1}(3) = \frac{log\ 3}{log\ 3} = 1[/tex]
