Question about inverse function for pre-calculus

Question

Determine whether f(x)= √x+8 has an inverse function. If it does, find the inverse function and state any restrictions on its domain. I already figured it does have an inverse function using the horizontal line test, but I don't know how to find the inverse function. 

Answer 1

I'm not sure whether the function is √(x + 8) or √x + 8, but the way you wrote it was the latter, so that's what I'll go with.

 

To find the inverse is pretty simple, with 4 steps:

 

1.  Change f(x) (or whatever the function name is) to "y".  That just makes it easier to work out the math.

 

In this case you'd get:

 

y = √x + 8

 

2.  Switch x and y.  Wherever you see y, put x.  And wherever you see x, put y.

 

x = √y + 8

 

3.  Solve the new equation for y.

 

x = √y + 8

x - 8 = √y

(x - 8)² = y

 

4.  Change y back to function notation.  The correct notation for the "inverse of f(x)" is "f^-1 (x)".  So that would give you:

 

f^-1(x) = (x - 8)²

 

 

Follow those same 4 steps every time you need to find the inverse of a function.

 

 

After that, remember that the domain of the inverse is the same as the range of the original function -- and the range of the inverse is the same as the domain of the original function.

 

The range of the original function was all real numbers greater than or equal to 8.  So that is the domain of the inverse.