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

Question

f(x)=3x-7
f(x)=x^2/(x^2-2)

Answer 1

To find the inverse of a function, we need to switch the x and y variables and solve for y.

For f(x) = 3x - 7:

Let y = 3x - 7
Swap x and y: x = 3y - 7
Solve for y: x + 7 = 3y
Divide both sides by 3: (x + 7) / 3 = y

The inverse function is f^-1(x) = (x + 7) / 3.

For f(x) = x^2 / (x^2 - 2):

Let y = x^2 / (x^2 - 2)
Swap x and y: x = y^2 / (y^2 - 2)
Multiply both sides by (y^2 - 2): x(y^2 - 2) = y^2
Expand: xy^2 - 2x = y^2
Rearrange terms: y^2 - xy^2 = 2x
Factor out y^2: y^2(1 - x) = 2x
Divide both sides by (1 - x): y^2 = 2x / (1 - x)
Take the square root of both sides: y = ±√[2x / (1 - x)]

The inverse function has two branches: f^-1(x) = ±√[2x / (1 - x)].