Question

Consider the functions f (x )equals negative 9 x plus 3 and g (x )equals negative one ninth (x minus 3 ). ​(a) Find​ f(g(x)). ​(b) Find​ g(f(x)). ​(c) Determine whether the functions f and g are inverses of each other.

Answer 1

Answer:

$(a) f(g(x)) = x\\(b) g(f(x)) = x$

(c) Yes, the functions f and g are inverses of each other.

Step-by-step explanation:

Given the functions:

$f(x) = -9x+3\\g(x) = -\dfrac{1}{9}(x-3)$

(a) $f(g(x))=?$

$put\ x = -\dfrac{1}{9}(x-3)\ in\ (-9x+3):$

$f(g(x))= -9(-\dfrac{1}{9}(x-3)) +3\\\Rightarrow (-\dfrac{-9}{9}(x-3)) +3\\\Rightarrow (\dfrac{9}{9}(x-3)) +3\\\Rightarrow 1(x-3) +3\\\Rightarrow x-3 +3\\\Rightarrow x\\\Rightarrow f(g(x) )=x$

(b) $g(f(x))=?$

$put\ x = (-9x+3)\ in\ -\dfrac{1}{9}(x-3):$

$f(g(x))= (-\dfrac{1}{9}((-9x+3)-3))\\\Rightarrow (-\dfrac{1}{9}(-9x+3-3))\\\Rightarrow (-\dfrac{1}{9}(-9x))\\\Rightarrow (-\dfrac{-9}{9}x)\\\Rightarrow g(f(x))=x$

(c) Yes, f and g are the inverse functions of each other.

As per the property of inverse function:

If $f^{-1}(x)$ is the inverse of $f(x)$ then:

$f(f^{-1}(x)) = x$

And here, we have the following as true:

$f(g(x)) = x\\ g(f(x)) = x$

$\therefore$ f and g are inverse functions of each other.