Question

Check whether f: R – {3} → R defined as f(x) = `(x-2)/(x-3)` is onto or not.

Answer 1

y = f(x)

y = `(x-2)/(x-3)`

y(x – 3) = (x – 2)

xy – 3y = x – 2

xy – x = 3y – 2

x (y – 1) = 3y – 2

x = `(3y-2)/(y-1)`

y ≠ 1

Range R – {1}

∴ The range of the function is the set of all possible y values, which is R – {1}. Since the range (R – {1}) is not equal to the codomain (R), the function is not onto.