Use the definition of inverses to determine whether f and g are inverses.
-$f ( x ) = 6 x ^ { 2 } + 1 , x \geq 0 ; \quad g ( x ) = \sqrt { \frac { x - 1 } { 6 } } , x \geq 1$
A) No
B) Yes

Question

Quizplus · Accepted Answer

To determine if f and g are inverses, we need to check if $ f(g(x)) = x $ and $ g(f(x)) = x $. For $ f(g(x)) $, substituting $ g(x) $ into $ f(x) $ gives $ f(\sqrt{\frac{x-1}{6}}) = 6(\sqrt{\frac{x-1}{6}})^2 + 1 = x $. For $ g(f(x)) $, substituting $ f(x) $ into $ g(x) $ gives $ g(6x^2 + 1) = \sqrt{\frac{6x^2 + 1 - 1}{6}} = x $. Both compositions equal x, confirming that f and g are inverses.

Use the Definition of Inverses to Determine Whether F and G