What is the inverse of f(x)=arctan(sqrt(4+e^{-8x)}) ?

The inverse of f(x)=arctan(sqrt(4+e^{-8x)}) is -(ln(tan^2(x)-4))/8

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inverse of f(x)=arctan(sqrt(4+e^{-8x)})

inverse

f (x) = arctan (4 + e^{- 8 x})

- \frac{ln ( tan ^{2} ( x ) - 4 )}{8}

Solution steps

arctan (a) = b \Rightarrow a = tan (b)

4 + e^{- 8 x} = tan (y)

Solve for

x, 4 + e^{- 8 x} = tan (y)

- \frac{ln ( tan ^{2} ( y ) - 4 )}{8}

Substitute

y = x

- \frac{ln ( tan ^{2} ( x ) - 4 )}{8}

\frac{2 ^{x} - 2 ^{- x}}{2}

f (x) = 3 (x - 1)^{2} + 2

ln (5 x + 3)

f (x) = 10 x - 4

What is the inverse of f(x)=arctan(sqrt(4+e^{-8x)}) ?
The inverse of f(x)=arctan(sqrt(4+e^{-8x)}) is -(ln(tan^2(x)-4))/8