Question

d. If f¹(x+2) =X-1/x+1x not equal to 1 then find f(x) and f¹(4).​

Answer 1

I assume by f¹, you actually mean f⁻¹ as in the inverse of f. I also assume you are asked to find f(x) (as in the inverse of f⁻¹) and f⁻¹(4).

Given that

$f^{-1}(x+2) = \dfrac{x-1}{x+1}$

with x ≠ 1, we can find f⁻¹(x) by replacing x + 2 with x :

$f^{-1}(x + 2) = \dfrac{x-1}{x+1} = \dfrac{(x+2) - 3}{(x + 2) - 1} \implies f^{-1}(x) = \dfrac{x-3}{x-1}$

Then when x = 4, we have

$f^{-1}(4) = \dfrac{4-3}{4-1} = \dfrac13$

Of course, we also could have just substituted x = 2 into the definition of f⁻¹(x + 2) :

$f^{-1}(4) = f^{-1}(2+2) = \dfrac{2-1}{2+1} = \dfrac13$

To find f(x), we fall back to the definition of an inverse function:

$f^{-1}\left(f(x)\right) = x$

Then by definition of f⁻¹, we have

$f^{-1}\left(f(x)\right) = \dfrac{f(x)-3}{f(x)-1} = x$

Solve for f :

$f(x) - 3 = x (f(x) - 1)$

$f(x) - 3 = x f(x) - x$

$f(x) - x f(x) = 3 - x$

$(1 - x) f(x) = 3-x$

$f(x) = \dfrac{3-x}{1-x} = \dfrac{x-1}{x-3}$