Question 1129946
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The usual way that students are taught to find the inverse of a function y=f(x) is to switch the x and y and solve for the new y.  For this example it might look like this:<br>
{{{x = 6y+1}}}
{{{x-1 = 6y}}}
{{{y = (x-1)/6}}}<br>
Tutor @josgarithmetic shows a different way of finding the inverse of a relatively simple function like this -- by solving the equation<br>
(f(f^(-1)(x)) = x:<br>
{{{f(f^(-1)(x)) = x}}}
{{{6(f^(-1)(x))+1 = x}}}
{{{6(f^(-1)(x)) = x-1}}}
{{{f^(-1)(x) = (x-1)/6}}}<br>
That is a very good formal mathematical way to find the inverse of a simple function, based on the idea that f(f^(-1)(x)) = x.<br>
If you don't need a formal derivation, you can do exactly the same thing as this second method without the formal mathematics, using the idea that the inverse function "gets you back where you started".<br>
In this example, the given function takes an input value and does two things:
(1) multiply by 6; and
(2) add 1.<br>
The inverse function, to get you back where you started, has to do the opposite operations in the opposite order:
(1) subtract 1; and
(2) divide by 6.<br>
Using this informal method, we quickly see that the inverse function is<br>
{{{f^(-1)(x) = (x-1)/6}}}<br>
So now you have at your disposal three different ways to find the inverse of a relatively simple function:
(1) switch x and y and solve for the new y
(2) solve the equation f(f^(-1)(x)) = x for f^(-1)(x)
(3) do the opposite operations in the opposite order<br>
Try using all three and find which one(s) work best for you....