Understanding Inverse Functions: How to Find f^-1(y)

[sonya]

if ur given a function f(x) and ur asked to find f^-1(y)...r u supposed 2 solve ur original eqn for y and then take the inverse of that? or isn't that just the same thing neways?...

 

[HallsofIvy]

What language precisely is that?

Anyway-

I don't know exactly how you have been taught to find inverse functions- there are several ways to arrive at the same result.

The way I like is this: Swap x and y.

Yes, that's it: If f(x)= y then f^-1(y)= x.

If f(10)= 0 then f^-1(0)= 10.

If f(x) is given by y= f(x)= 3x- 4 then the inverse function is given by x= 3y- 4.

Oh, there is one tiny other thing you might want to do:
Since we prefer to write f(x)= ... or f^-1(x)= ...,
you might want to solve for y!

Since x= 3y- 4, 3y= x+ 4 and y= f^-1(x)= (x+4)/3.

Notice the key point: what f(x) "does", f^-1(x) "undoes".
Where f(x) is "multiply by 3 then add 4", f^-1(x) says "subtract 4, then divide by 3". Each step is reversed ("add 4" instead of "subtract 4" and "divide by 3" instead of "multiply by 3") and the order is also reversed. Of course: when I go to work in the morning, I put on my shoes, then go out the door, then lock the door behind me. When I come home in the evening, I first UNlock the door, then go in through the door, then take off my shoes. Each operation is reversed and the order is reversed.

Of course, you can't always "solve" for the inverse function.

If f(x)= e^x then f^-1(x)= ln(x) because that is the way ln is DEFINED- as the inverse function to e^x.

 

[sonya]

Originally posted by HallsofIvy

Since x= 3y- 4, 3y= x+ 4 and y= f^-1(x)= (x+4)/3.

ok..i think i understand now but a quick question
where x = 3y -4 that can also be called f^-1(y) rite?

 

[sonya]

o..never mind that question...i get it now
thx a lot!