Two functions f and g are defined on the set R of real numbers by \(f : x \to 2x - 1\) and \(g : x \to x^{2} + 1\). Find the value of \(f^{-1} \circ g(3)\).

A.

12

B.

11

C.

\(\frac{11}{2}\)

D.

\(\frac{9}{2}\)

Correct answer is C

\(f(x) = 2x - 1; g(x) = x^{2} + 1\)

\(f(x) = 2x - 1\)

\(f(y) = 2y - 1 \)

\(x = 2y - 1 \implies 2y = x + 1\)

\(y = \frac{x + 1}{2}\)

\(g(3) = 3^{2} + 1 = 10\)

\(f^{-1}(10) = \frac{10 + 1}{2} = \frac{11}{2}\)