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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.
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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