The function f(x)=ex−e−x2 is one-to-one. Find the formula for f−1. [Hint: Solve for x in the equation y=ex−e−x2. You may need to convert to a quadratic in form equation, which you may then solve using the quadratic formula.]

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Lauren Fuller · Accepted Answer

Step 1  To find the inverse of any function y=f(x), we need to replace x by y and y by x so as to find a function y=g(x).  Here, we get:  f−1(x)=g(x)  Step 2  The inverse to the given function can be evaluated as shown below:  y=ex−e−x2  1) (x)=e(y)−e−(y)2 [Replacing x by y and y by x]  x2=(ey−e−y2)2  x2=14(e2y−2+e−2y)  x2+1=14(e2y−2+e−2y)+44  x2+1=14(e2y+2+e−2y)  x2+q=14(ey+e−y)2x2+1=14(2ey−(ey−e−y))2  x2+1=12(2ey−(ey−e−y))  x2+1=ey−(ey−e−y)2  x2+1=ey−x [Using equation (i)]  ey=x+x2+1  y=ln⁡(x+x2+1)  Thus we get,  f−1(x)=ln⁡(x+x2+1)

The function f(x)=\frac{e^{x}-e^{-x}}{2} is one-to-one. Find the formula for f^{-1}.

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