Decide whether or not the functions are inverses of each other.
-$$f ( x ) = ( x - 6 ) ^ { 2 } , x \geq 6 ; g ( x ) = \sqrt { x + 6 }$$
A) Yes
B) No

Question

Quizplus · Accepted Answer

The functions $f(x) = (x - 6)^2, x \geq 6$ and $g(x) = \sqrt{x + 6}$ are not inverses of each other because when you apply $g(f(x))$, you get $\sqrt{(x-6)^2 + 6}$, which simplifies to $\sqrt{x^2 - 12x + 36 + 6} = \sqrt{x^2 - 12x + 42}$, not $x$. Similarly, applying $f(g(x))$ does not simplify back to $x$. The domain restriction $x \geq 6$ for $f(x)$ ensures that the square root in $g(x)$ is defined, but it does not guarantee that the two functions are inverses of each other.

Decide Whether or Not the Functions Are Inverses of Each f(x)=(x−6)2,x≥6;g(x)=x+6f ( x ) = ( x - 6 ) ^ { 2 } , x \geq 6 ; g ( x ) = \sqrt { x + 6 }f(x)=(x−6)2,x≥6;g(x)=x+6​

Decide Whether or Not the Functions Are Inverses of Each $f ( x ) = ( x - 6 ) ^ { 2 } , x \geq 6 ; g ( x ) = \sqrt { x + 6 }$