SOLUTION: If f(x)=square root of x-1 find f-1(x)

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Question 112746: If f(x)=square root of x-1 find f-1(x)
Found 2 solutions by mathnard, solver91311:
Answer by mathnard(10) About Me  (Show Source):
You can put this solution on YOUR website!
This is an inverse functions problem. f(x) is read f of x, not f times x and is just a fancy way of saying y. so really what they are saying is:
y=sqrt%28x-1%29
To find an inverse, which is what f%5E%28-1%29 means, you switch the x and y and then solve for the new y:

x=sqrt%28y-1%29
square both sides:
%28x%29%5E2=%28sqrt%28y-1%29%29%5E2
so
x%5E2=y-1
add 1 to both sides
x%5E2%2B1=y

This means that the inverse function is:
f%5E-1%28x%29=x%5E2%2B1

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
I think you mean: If f%28x%29=sqrt%28x-1%29, then find f%5E%28-1%29%28x%29.

The first thing you have to do is determine whether f%5E%28-1%29%28x%29 exists. There is a pretty simple test that will tell you. It is called the horizontal line test.

Step 1: Graph the original function.

graph%28400%2C400%2C-2%2C5%2C-2%2C5%2Csqrt%28x-1%29%29

Step 2: If ANY possible horizontal line intersects the graph in more than one place, then we know that f%5E%28-1%29%28x%29 is NOT a function, i.e. f does NOT have an inverse. Otherwise, f%5E%28-1%29%28x%29 does exist and f does have an inverse. For this example, there is no horizontal line that intersects the graph more than once, so we know that f has an inverse. Now, and only now, can we set about finding it.

Procedure to find the inverse of a one-to-one function (one that has an inverse):

Step 1: Replace f(x) with y.

y=sqrt%28x-1%29

Step 2: Swap the positions of the x and y variables with each other.

x=sqrt%28y-1%29

Step 3: Rearrange the equation so that it again is showing y as a function of x, i.e. solve for y.

x=sqrt%28y-1%29
x%5E2=y-1
y=x%5E2%2B1


Ok, Super-Double-Plus Extra Credit. If f%28x%29=x%5E2%2B1, does f%5E%28-1%29%28x%29 exist?

Hope this helps
John