The required value of function g(x) = [tex]\frac{14}{x+3}[/tex], [tex]((g^{-1} . g )(4))[/tex] is 4 .
Given,
The function g(x) = [tex]\frac{14}{x+3}[/tex],
We have to find [tex](g^{-1} . g )(4))[/tex]
According to the question,
Two functions f and g are inverse functions if fog(x) = x and gof (x) = x for all values of x in the domain of f and g.
g(x) = [tex]\frac{14}{x+3}[/tex]
at x = 4
[tex]= \frac{14}{4+3} \\\\= \frac{14}{7} \\\\g(4) = 2[/tex]
Then,
[tex]= (g^{-1} . g )(4) = (g^{-1} .(2))[/tex]
Now, Let g(X) = y
[tex]y = \frac{14}{x+3}[/tex]
Cross multiplication,
y ( x +3) = 14
yx + 3y = 14
yx = 14 - 3y
x = [tex]\frac{14-3y}{y}[/tex]
Now, [tex]= (g^{-1} .(x))=\frac{14-3y}{y}[/tex]
At x = 2
= [tex]\frac{14 - 3 (2)}{2}[/tex]
[tex]= \frac{14-6}{2} \\\\= \frac{8}{2}[/tex]
= 4
Hence, The required value of [tex]((g^{-1} . g )(4))[/tex] is 4 .
For the more information about the Function click the link given below.
https://brainly.com/question/14418346
