Find an equation for the inverse of each of these functions:?

Question

a.) g(x) = x^2-7, x greater than or equal to 0

b.) h(x) = x+3 / 2x-4

Answer 1

a) g(x) = x^2 - 7, x >= 0

First, make g(x) into y.

y = x^2 - 7

Swap the x and y terms, and solve for y.

x = y^2 - 7

x - 7 = y^2

Therefore, y = +/- sqrt(x - 7).
However, since x >= 0 in the original function, it follows that its inverse requires y >= 0. Therefore, we take the positive result, and the inverse is

g^(-1)(x) = sqrt(x - 7).

b) h(x) = (x + 3)/(2x - 4)

First, make h(x) into y.

y = (x + 3)/(2x - 4)

Swap the x and y terms.

x = (y + 3)/(2y - 4)

Multiply both sides by (2y - 4)

x(2y - 4) = y + 3

Expand the left hand side.

2xy - 4x = y + 3

Move everything with a y to the left hand side; everything else goes to the right hand side.

2xy - y = 4x + 3

Factor y,

y(2x - 1) = 4x + 3

y = [4x + 3]/[2x - 1]

Therefore,

h^(-1)(x) = [4x + 3]/[2x - 1]