Find the inverse functions (f^-1(x)) of the functions below

Question

f(x)=x^3+3
f(x)=(x+1)/(x-2) ; x≠2

Answer 1

To find the inverse function, we want to solve for x in terms of y.

1) f(x) = x^3 + 3

Let y = x^3 + 3

Swap x and y:

x = y^3 + 3

Now solve for y:

x - 3 = y^3

Take the cube root of both sides:

∛(x - 3) = y

So, the inverse function is f^(-1)(x) = ∛(x - 3).

2) f(x) = (x + 1)/(x - 2), x ≠ 2

Let y = (x + 1)/(x - 2)

Swap x and y:

x = (y + 1)/(y - 2)

Now solve for y:

xy - 2x = y + 1

xy - y = 2x + 1

Factor out y:

y(x - 1) = 2x + 1

Divide both sides by (x - 1):

y = (2x + 1)/(x - 1)

So, the inverse function is f^(-1)(x) = (2x + 1)/(x - 1), x ≠ 1.