Find f(x) when Inverse f(x) is given.

Question

f^-1(1/(x+1)) = 3x+4, then find f(x) = ?

Answer 1

f^-1\((\frac{1}{x+1})=3x+4\)

\(\therefore\) f^-1\(\left(\cfrac{1}{\frac{1-x}x+1}\right)\) = 3\((\frac{1-x}x+4)\)

⇒ f^(-1)(x) = 3\((\frac{1-x}x)+4\)---(1)

Let f^-1 = y

⇒ f(y) = x

\(\therefore\) From (1), we get

y = \(3(\frac{1-x}x)+4\)

⇒ \(\frac{x(y-4)}3=1-x\) 

⇒ x + x/3(y - 4) = 1

⇒ x (1 + \(\frac{y-4}3\)) = 1

⇒ x \((\frac{3+y-4}3)=1\)

⇒ x (\(\frac{3+y-4}3\)) = 1

⇒ x = 3/y-1

⇒ f(y) = 3/y-1

\(\therefore\) f(x) = 3/x-1