Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. (5 points)

f(x) = x2 - 3 and g(x) = square root of quantity three plus x

Answer 1

f(g(x)) = g(f(x)) = x and f and g are the inverses of each other.

Explanation:

Here, the given functions are:

`f(x) = x^2 - 3, g(x) = \sqrt{({3+x)} }`

To Show: f (g(x)) = g (f (x))

(1) f (g(x))

Here, by the composite function:

`f (g(x)) = f (√(3+x) )  = √((3+x)) ^2 - 3  =  (3 + x) - 3  =  x`

⇒ f (g(x)) = x

(2) g (f(x))

Here, by the composite function:

`g(f(x)) = g(x^2 -3)   = √(3 +(x^2 -3) )  = √(x^2)   = x`

⇒ g (f(x)) = x

Hence, f(g(x)) = g(f(x)) = x

⇒ f and g are the inverses of each other.