Question

I need help please, been stuck on this question.

Answer 1

D)
`(1)/(6)`

g¹(1) =
`(1)/(6)`

The inverse of the function
`g(x) = \frac{x^{(1)/(3) }-1 }{2}`

Explanation:

Step(i):-

Given that f(x) = (2x+1)³

Let y = (2x+1)³

`y^{(1)/(3) } =2x+1`

`2x = y^{(1)/(3) } -1`

`x = \frac{y^{(1)/(3) }-1 }{2}`

Step(ii):-

y = f(x) ⇒ x = f⁻¹ (y)

⇒
`f^(-1) (y) = \frac{y^{(1)/(3) }-1 }{2}`

`f^(-1) (x) = \frac{x^{(1)/(3) }-1 }{2}`

The inverse of the given function

`g(x) = \frac{x^{(1)/(3) }-1 }{2}`

Differentiating equation (i) with respective to 'x', we get

`g^(l) (x) = (1)/(2) X (1)/(3) x^{(1)/(3) -1}`

`g^(l) (x) = (1)/(6) x^{(-2)/(3) }`

Final answer:-

Put x=1

`g^(l) (1) = (1)/(6) 1^{(-2)/(3) } = (1)/(6)`