Answer: f(g(x))=x and g(f(x)) = x
f⁻¹(x) = g(x) YES ARE INVERSES
f⁻¹(x) ≠ g(x) NOT INVERSES
Step-by-step explanation:
Inverse is when you swap the x's and y's and then solve for y.
If f⁻¹(x) = g(x), then they are inverses of each other.
Similarly, if g⁻¹(x) = f(x), they are inverses of each other.
NOTE: You can also use composition to determine if they are inverses --> If (fog)(x) = x, then they are inverses of each other.
[tex]f(x) = \dfrac{1}{x+4}-9\\\\\\\text{Swap the x's and y's. NOTE: f(x) is y}\\x=\dfrac{1}{y+4}-9\\\\\\\text{Add 9 to both sides}\\x+9=\dfrac{1}{y+4}\\\\\\\text{Flip the fractions}\\\dfrac{1}{x+9}=y+4\\\\\\\text{Subtract 4 from both sides}\\\dfrac{1}{x+9}-4=y\\\\\\\boxed{f^{-1}(x)=g(x)\text{ so f(x) and g(x) are inverses of each other}}[/tex]
[tex]f(x) = 3x+27\\\\\\\text{Swap the x's and y's NOTE f(x) is y}\\x=3y+27\\\\\\\text{Subtract 27 from both sides}\\x-27=3y\\\\\\\text{Divide everything by 3}\\\dfrac{1}{3}x-\dfrac{27}{3}=y\\\\\\\text{Simplify}\\\dfrac{1}{3}x-9=y\\\\\\\boxed{f^{-1}(x)\neq g(x)\text{ so f(x) and g(x) are NOT inverses of each other}}[/tex]
