Question

What are two methods you can use to tell if 2 functions are inverses of each other?

Answer 1

Method One

f(g(x)) = x

Method Two

g(f(x)) = x

So let's pick a pair of functions and try this out.

f(x) = x^2 + 1

g(x) =sqrt(x - 1)

Using Method 1

f(g(x)) = (g(x)^2 + 1              You put a g(x) wherever you see an x in f(x)

f(g(x)) = [sqrt(x - 1)}^2 + 1    Substitute the right side of g(x) on the right side of   f(x)

f(g(x)) = x - 1 + 1                   Expand and cancel

f(g(x) = x

Using Method 2

g(f(x)) = sqrt(f(x) - 1)             Put an f(x) wherever you see an x in g(x)

g(f(x)) = sqrt(x^2 + 1 - 1)       Substitute the value of f(x) in the g(x) equation

g(f(x)) = sqrt(x^2)                 The 1s cancel. Take the square root of x^2

g(f(x)) = x                              You get x which is what you need to get.  

So these two functions are the inverses of each other. Both methods confirm the results. A graph may help you to understand.

Notice how the red line (f(x) = x^2 + 1) is reflected across the green line to become the blue line (g(x) = sqrt(x - 1) ) That is another way to tell that 2 equations are inverses.

Note further that I have take the equations so that x in all three cases is ≥ 0