Can you please show steps because I'm really confused
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g(x) is just the hyperbolic cosine function evaluated at 2x: cosh(2x).
http://en.wikipedia.org/wiki/Hyperbolic_…
For now write z = 2x and y = g(x).
Then y = (e^z+e^-z)/2 ==> 2y = e^z + e^-z ==> e^2z -2ye^z + 1 = 0 ==> (e^z)^2 -2y(e^z) + 1 = 0.
This last equations is a quadratic in the variable e^z, so applying the quadratic formula gives:
e^z = y +/- sqrt(y^2-1) ==> 2x = z = ln(y+/-sqrt(y^2-1)) ==> x = ln(y +/- sqrt(y^2-1))/2.
Substituting x for y in the above equation we have:
g-1(x) = in (x +/- sqrt(x^2-1))/2.
Since g(x) is always positive, we reject the value ln(x - sqrt(x^2-1))/2 (which is negative for large x), and have:
g-1(x) = in (x +/- sqrt(x^2-1))/2.
http://en.wikipedia.org/wiki/Hyperbolic_…
For now write z = 2x and y = g(x).
Then y = (e^z+e^-z)/2 ==> 2y = e^z + e^-z ==> e^2z -2ye^z + 1 = 0 ==> (e^z)^2 -2y(e^z) + 1 = 0.
This last equations is a quadratic in the variable e^z, so applying the quadratic formula gives:
e^z = y +/- sqrt(y^2-1) ==> 2x = z = ln(y+/-sqrt(y^2-1)) ==> x = ln(y +/- sqrt(y^2-1))/2.
Substituting x for y in the above equation we have:
g-1(x) = in (x +/- sqrt(x^2-1))/2.
Since g(x) is always positive, we reject the value ln(x - sqrt(x^2-1))/2 (which is negative for large x), and have:
g-1(x) = in (x +/- sqrt(x^2-1))/2.