What is the inverse of f(x)= 1/(2-sin(3x)) ?

The inverse of f(x)= 1/(2-sin(3x)) is (arcsin(-(1-2x)/x))/(3)

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inverse of f(x)= 1/(2-sin(3x))

inverse

f (x) = \frac{1}{2 - sin ( 3 x )}

\frac{arcsin ( - \frac{1 - 2 x}{x} )}{3}

Solution steps

Solve for

sin (3 x), y = \frac{1}{2 - sin ( 3 x )}

sin (3 x) = - \frac{1 - 2 y}{y}

sin (a) = b \Rightarrow a = arcsin (b)

3 x = arcsin (- \frac{1 - 2 y}{y})

Solve for

x, 3 x = arcsin (- \frac{1 - 2 y}{y})

\frac{arcsin ( - \frac{1 - 2 y}{y} )}{3}

Substitute

y = x

\frac{arcsin ( - \frac{1 - 2 x}{x} )}{3}

f (x) = 12 - 2 x - \frac{3}{2}

y = x^{2} - 4 x, x \leq 0

f (x) = 2 \cdot 1 0^{x}

Q (x) = 45 - 0.45 x

What is the inverse of f(x)= 1/(2-sin(3x)) ?
The inverse of f(x)= 1/(2-sin(3x)) is (arcsin(-(1-2x)/x))/(3)