Answer:
D) [tex]\frac{1}{6}[/tex]
g¹(1) = [tex]\frac{1}{6}[/tex]
The inverse of the function [tex]g(x) = \frac{x^{\frac{1}{3} }-1 }{2}[/tex]
Step-by-step explanation:
Step(i):-
Given that f(x) = (2x+1)³
Let y = (2x+1)³
[tex]y^{\frac{1}{3} } =2x+1[/tex]
[tex]2x = y^{\frac{1}{3} } -1[/tex]
[tex]x = \frac{y^{\frac{1}{3} }-1 }{2}[/tex]
Step(ii):-
y = f(x) ⇒ x = f⁻¹ (y)
⇒ [tex]f^{-1} (y) = \frac{y^{\frac{1}{3} }-1 }{2}[/tex]
[tex]f^{-1} (x) = \frac{x^{\frac{1}{3} }-1 }{2}[/tex]
The inverse of the given function
[tex]g(x) = \frac{x^{\frac{1}{3} }-1 }{2}[/tex]
Differentiating equation (i) with respective to 'x', we get
[tex]g^{l} (x) = \frac{1}{2} X \frac{1}{3} x^{\frac{1}{3} -1}[/tex]
[tex]g^{l} (x) = \frac{1}{6} x^{\frac{-2}{3} }[/tex]
Final answer:-
Put x=1
[tex]g^{l} (1) = \frac{1}{6} 1^{\frac{-2}{3} } = \frac{1}{6}[/tex]
