# Find the inverse of the function f(x)=4+e^-(3x)Find the inverse of the function f(x)=4+e^-(3x). FInd the domain, range, and asymptotes of each function. Graph both functions on the same coordinate...

Find the inverse of the function f(x)=4+e^-(3x)

Find the inverse of the function f(x)=4+e^-(3x). FInd the domain, range, and asymptotes of each function. Graph both functions on the same coordinate plane.

f(x)=4+e^-(3x)

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`y= 4+e^-(3x)`

If y=f(x) inverse function will be x=f(y)

`y-4 = e^(-3x)`

`ln(y-4) = ln(e^(-3x))`

`ln(y-4) = -3x`

`(-1/3)ln(y-4) = x`

`ln[1/(y-4)^(1/3)]= x`

So if we interchange x and y as a normal function inverse function can be expressed as;

y = `ln[1/(x-4)^(1/3)]`

The graphs are shown below.

The black graph shows the initial function and the red graph shows the inverse function.

For `y= 4+e^-(3x)`

Domain: `x in Z`

Range : `y >4`

For y= `ln[1/(x-4)^(1/3)]`

Domain : `xgt4`

Range : y< `4+e^(-3x)`

To find the inverse of the function `y=4+e^{-3x}`, we need to interchange the x and the y and then solve for y. This means that we are solving for y in

`x=4+e^{-3y}`

` ` `e^{-3y}=x-4` now take logarithms

`-3y=ln(x-4)`

`y=-1/3 ln(x-4)`

The original function has a domain of all real numbers, range of all real numbers `y>4`, and a horizontal asymptote `y=4`.

The inverse function has domain of all real numbers `x>4`, range of all real numbers and a vertical asymptote `x=4`.