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Tanzania [10]
3 years ago
15

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.
Mathematics
1 answer:
zlopas [31]3 years ago
7 0

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.

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