Suppose D(f) = {x| x ∈ R, x ≠ 3} and f(x) =(x+2) /(x-3). What does define(s) inverse (f−1)?

Answer 1

x = (y+2)/(y-3)
xy - 3x = y + 2
y(x-1) = 3x+2
g(x) = f^-1(x) = y = (3x+2)/(x-1)

check: f(x) and g(x) are inverses if (f◦g)(x) = (g◦f)(x) = x
g(f) = (3f+2)/(f-1) = (3(x+2)/(x-3)+2)/((x+2)/(x-3) - 1)
= (3x+6+2x-6)/(x+2-x+3)
= 5x/5
= x

f(g) = (g+2)/(g-3) = ((3x+2)/(x-1) + 2)/((3x+2)/(x-1) - 3)
= (3x+2+2x-2)/(3x+2-3x+3)
= 5x/5
= x

Answer 2

Thanks oobleck