The required value of function g(x) = \(\frac{14}{x+3}\), \(((g^{-1} . g )(4))\) is 4 .
Given,
The function g(x) = \(\frac{14}{x+3}\),
We have to find \((g^{-1} . g )(4))\)
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) = \(\frac{14}{x+3}\)
at x = 4
\(= \frac{14}{4+3} \\\\= \frac{14}{7} \\\\g(4) = 2\)
Then,
\(= (g^{-1} . g )(4) = (g^{-1} .(2))\)
Now, Let g(X) = y
\(y = \frac{14}{x+3}\)
Cross multiplication,
y ( x +3) = 14
yx + 3y = 14
yx = 14 - 3y
x = \(\frac{14-3y}{y}\)
Now, \(= (g^{-1} .(x))=\frac{14-3y}{y}\)
At x = 2
= \(\frac{14 - 3 (2)}{2}\)
\(= \frac{14-6}{2} \\\\= \frac{8}{2}\)
= 4
Hence, The required value of \(((g^{-1} . g )(4))\) is 4 .
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