How would you find out if g(x)= -x+1 is the inverse of f(x)= x2-2?

Question

I just need to find out how you would check to see if they are inverses... It's for my math exam review, and I forgot (we learned it a while back) ... and I need to study.

Answer 1

Two functions, for example f and g, are inverses of each other if f(g(x))=g(f(x))=x. Letting f=x2-2 and g=-x+1:

f(g(x)) = f(-x+1) = (-x+1)2-2
= -2x+2-2 = -2x

g(f(x)) = g(x2-2)
= -(x2-2)+1=-x2+3

As you can see, f and g are not inverses of each other.

Answer 2

Set the function equal to another variable, say "y": f(x) = y y = (x^2 - 4) / (2x^2) Now solve this for x in terms of y. You'll get a new function in terms of y. This function will be the inverse of f(x). The easiest way to do this is to probably just multiply both sides by 2x^2 and combine like terms. You should eventually get: (2y - 1)*x^2 = 4 x^2 = 4 / (2y-1) x = sqrt(4 / (2y-1)) etc. So the inverse function of f(x), let's call it g(y), is: g(y) = (2 * sqrt(2y-1)) / (2y-1)

Answer 3

Let y = g(x) = - x + 1
To find the inverse, solve y for x:
y = - x + 1
-(y - 1) = x
Thus, g(x)^(-1) = -(y - 1), which is not equal to f(x).

Answer 4

good question. i actually wrote it down because my algebra midterm is on tuesday and i should really start remembering how to do these...