What is the inverse of the function f(x) = (e^3x)*14 - 21

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The function given is f(x) = (e^3x)*14 - 21. We need to find the inverse of the function.

Let f(x) = y = (e^3x)*14 - 21

isolate x in terms of y

y = (e^3x)*14 - 21

=> y + 21 = (e^3x)*14

=> (y + 21)/14 = e^3x

take the logarithm to the base e of both the sides

=> ln [(y + 21)/14] = 3x*ln e

=> ln (y + 21) - ln 14 = 3x

=> x = [ln (y + 21) - ln 14]/3

interchange x and y

=> y = [ln (x + 21) - ln 14]/3

**The inverse of the given function is f(x) = [ln (x + 21) - ln 14]/3**

We'll suggest another method of getting the inverse function.

We know that the product of derivative and it's inverse is 1.

f'(x)*[f^-1(x)]' =1

We'll divide both sides by f'(x):

[f^-1(x)]' = 1/f'(x)

We'll calculate the integral of functions both sides:

Int [f^-1(x)]'dx = Int dx/f'(x)

We'll differentiate the function f(x):

f'(x) = [(e^3x)*14 - 21]'

f'(x) = 42*e^3x

f^-1(x) = Int dx/42*e^3x

Int dx/42*e^3x = (1/42)*Int e^-3xdx

Int dx/42*e^3x = -e^-3x/3*42

Int dx/42*e^3x = -1/126*e^3x

**Therefore, the inverse function is: f^-1(x) = -1/126*e^3x.**

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