Question

C. If f(4x+5)=12x+18 then find f¹(x).​

Answer 1

If f¹ is supposed to say f⁻¹ : rewrite f(4x + 5) as an "obvious" function of 4x + 5, then replace 4x + 5 with x. By "obvious", I mean make it clear how 4x + 5 is the argument to f.

`f(4x+5) = 12x+18 = 3 (4x + 6) = 3 (4x + 5 + 1) = 3 (4x + 5) + 3`

Then swapping out 4x + 5 for x gives

`f(x) = 3x + 3`

The inverse of f(x), if it exists, is a function f⁻¹(x) such that

`f\left(f^(-1)(x)\right) = x`

Evaluate f at f⁻¹ and solve for f⁻¹ :

`f\left(f^(-1)(x)\right) = 3 f^(-1)(x) + 3 = x \implies \boxed{f^(-1)(x) = \frac{x-3}3}`

If f¹ instead means f' (as in the first derivative of f) : earlier we found f(x) = 3x + 3, and differentiating this is trivial.

`f(x) = 3x+3 \implies f'(x) = 3`