Pre calculus 4b. The function g is defined as g(x) = InX, X E R^tThe graph of y = g(x) and the graph of y = f^-1 (x) intersect at the point P.Find the exact coordinates of P.

[tex]\begin{gathered} \ln y-1=3x+1-1 \\ \text{Then} \\ \ln y-1=3x \\ \text{Divide both sides by 3, we have } \\ \frac{\ln y-1}{3}=\frac{3x}{3} \end{gathered}[/tex]

Then

[tex]x=\frac{\ln y-1}{3}[/tex]

Then the inverse of the function becomes

[tex]f^{-1}(x)=\frac{\ln x-1}{3}[/tex]

Hence

The inverse of the function, f(x) is

f¹(x)=ln x - 1 / 3

The Domian of a function are the set of the input value of for which the function is defined or real.

Hence

[tex]\begin{gathered} \text{The domain is } \\ (0,\infty) \end{gathered}[/tex]

Therefore

The domain fo the inverse of f(x) is (0, ∞)

Part B

The function g(x) is given as

[tex]g(x)=\ln x[/tex][tex]f^{-1}(x)=\frac{\ln x-1}{3}[/tex]

The point of intersection is the point where

[tex]f^{-1}(x)=g(x)[/tex]

Then, we have

[tex]\begin{gathered} lnx=\: \frac{\ln\:x-1}{3} \\ \text{Multiply both sides by 3} \\ 3\text{ lnx=lnx-1} \end{gathered}[/tex]

Then, subtract both sides by ln x

[tex]\begin{gathered} 3\ln x-\ln x=\ln x-\ln x-1 \\ 2\ln x=-1 \end{gathered}[/tex]

Divide both sides with 2 we have

[tex]\begin{gathered} \frac{2\ln x}{2}=-\frac{1}{2} \\ \text{Then} \\ \ln x=-\frac{1}{2} \end{gathered}[/tex]

Then, take the exponent of both sides, we have

[tex]\begin{gathered} e^{\ln x}=e^{-\frac{1}{2}} \\ x=e^{-\frac{1}{2}} \end{gathered}[/tex]

Then

[tex]\begin{gathered} x=\frac{1}{e^{\frac{1}{2}}}=\frac{1}{\sqrt[]{e}} \\ \text{Hence } \\ x=\frac{1}{\sqrt[]{e}} \end{gathered}[/tex]

Then substitute the value of x into g(x) to find y, we have \

From the question,

[tex]\begin{gathered} y=g(x) \\ \text{and } \\ g(x)=\ln x \\ \text{Then} \\ y=\ln x \\ \text{ since x=e}^{-\frac{1}{2}} \\ \text{Then} \\ y=\ln e^{-\frac{1}{2}}=-\frac{1}{2} \end{gathered}[/tex]

Therefore the point of intersection P is

[tex](\frac{1}{\sqrt[]{e}},-\frac{1}{2})[/tex]

Consider the graph below

Therefore point P is (0.607, -0.5)