Answer:
[tex](a) f(g(x)) = x\\(b) g(f(x)) = x[/tex]
(c) Yes, the functions f and g are inverses of each other.
Step-by-step explanation:
Given the functions:
[tex]f(x) = -9x+3\\g(x) = -\dfrac{1}{9}(x-3)[/tex]
(a) [tex]f(g(x))=?[/tex]
[tex]put\ x = -\dfrac{1}{9}(x-3)\ in\ (-9x+3):[/tex]
[tex]f(g(x))= -9(-\dfrac{1}{9}(x-3)) +3\\\Rightarrow (-\dfrac{-9}{9}(x-3)) +3\\\Rightarrow (\dfrac{9}{9}(x-3)) +3\\\Rightarrow 1(x-3) +3\\\Rightarrow x-3 +3\\\Rightarrow x\\\Rightarrow f(g(x) )=x[/tex]
(b) [tex]g(f(x))=?[/tex]
[tex]put\ x = (-9x+3)\ in\ -\dfrac{1}{9}(x-3):[/tex]
[tex]f(g(x))= (-\dfrac{1}{9}((-9x+3)-3))\\\Rightarrow (-\dfrac{1}{9}(-9x+3-3))\\\Rightarrow (-\dfrac{1}{9}(-9x))\\\Rightarrow (-\dfrac{-9}{9}x)\\\Rightarrow g(f(x))=x[/tex]
(c) Yes, f and g are the inverse functions of each other.
As per the property of inverse function:
If [tex]f^{-1}(x)[/tex] is the inverse of [tex]f(x)[/tex] then:
[tex]f(f^{-1}(x)) = x[/tex]
And here, we have the following as true:
[tex]f(g(x)) = x\\ g(f(x)) = x[/tex]
[tex]\therefore[/tex] f and g are inverse functions of each other.
