The graph of the direct function shows that it is not one-to-one. 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
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+9-y=0
or 5x^2+(9-y)x+10=0, using standard quadratic solution dormula
for the x<=sqrt(2) =1.4 the inverse is
for the x>sqrt(2) =1.4 the inverse is
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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