how to find the inverse of a onetoone function
Algebra

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an example that is given is f(x)= 5x+10/x+9
y=fx.pdf
The graph of the inverse function in give it is obtained by reflecting around the y=x axis(a line inclined at 45deg with resspect to the axis
The graph of the direct function shows that it is not onetoone. It can be made into one by separating it into
the x>2.6 and x<=2.6 (2.6 is the maximum of the inverted parabola as obtained from the graph)
For each of these 2 branches a single x=f^1(y) can be found as shown on the second graph. In the second graph the x is replaced by y (and is now the vertical axis). The inverse has 2 branches one above and one below the x=2.6 value.
A correction  the max of the function actually falls at x=sqrt(2)= 1.4 (not 2.6) please replace above 2.6 by 1.4
Good luck
one more comment  the value of the y on the first graph (y=fx) and the value of the x on the second should be multiplied by 10 (sorry, it did indicate that with the program I used to plot.) Thus, the max when x= 1.4 is y=23.
The analytical solutions are found as follows, we solve the equation 5x+10/x+9y=0
or 5x^2+(9y)x+10=0, using standard quadratic solution dormula
for the x<=sqrt(2) =1.4 the inverse is
x=(sqrt(y^218y119)+y9)/10
for the x>sqrt(2) =1.4 the inverse is
x=(sqrt(y^218y119)+y9)/10
Note that inverse at the point x=sqrt(2) we could have included in either branch. We chose arbitrarily the first one .
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