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

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

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inverse of f(x)= 1/2 sin(x/2)+1/2

inverse

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

2 arcsin (2 x - 1)

Solution steps

Solve for

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

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

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

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

Solve for

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

2 arcsin (2 y - 1)

Substitute

y = x

2 arcsin (2 x - 1)

f (x) = \frac{( 2 x - 5 ) ( 2 x + 5 )}{x ^{2}}

(- P, 0), (0, - R)

f (x) = \frac{7}{7 + x}

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

(9, - 8), (- 7, - 5)

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