how to find the inverse of a one-to-one function

Algebra
Tutor: None Selected Time limit: 1 Day

an example that is given is f(x)= 5x+10/x+9

May 3rd, 2015
Graph of the direct function
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

x=f-1(y).pdf

May 3rd, 2015

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.

May 3rd, 2015

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

May 3rd, 2015

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.

May 3rd, 2015

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

x=(sqrt(y^2-18y-119)+y-9)/10

for the x>sqrt(2) =1.4 the inverse is 

x=(-sqrt(y^2-18y-119)+y-9)/10

Note that inverse at the point x=sqrt(2) we could have included in either branch. We chose arbitrarily the first one .


May 3rd, 2015

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