Question

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

Answer 1

Answer:

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

Step-by-step 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:

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

⇒ f (g(x))  = x

(2) g (f(x))

Here, by the composite function:

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

⇒ g (f(x))  = x

Hence, f(g(x)) = g(f(x))  = x

⇒ f and g are the inverses of each other.