Show that f: R − {3} → R − {2}, f(x) = (2x + 1)/(x − 3) is one-one and onto.

1. 1 · Write what is given and what is to be found. Given: f(x) = (2x + 1)/(x − 3) from R − {3} to R − {2}. To show: f is one-one and onto.
2. 2 · Write the rules.
   • One-one: f(x₁) = f(x₂) ⇒ x₁ = x₂.
   • Onto: every y in R − {2} has some x in R − {3} with f(x) = y.
3. 3 · Prove one-one.
   • (2x₁ + 1)(x₂ − 3) = (2x₂ + 1)(x₁ − 3)
   • 2x₁x₂ − 6x₁ + x₂ − 3 = 2x₁x₂ − 6x₂ + x₁ − 3
   • −6x₁ + x₂ = −6x₂ + x₁, so 7x₂ = 7x₁ and x₁ = x₂
4. 4 · Solve y = f(x) for x.
   • y(x − 3) = 2x + 1
   • xy − 2x = 3y + 1, so x(y − 2) = 3y + 1
   • x = (3y + 1)/(y − 2), which is real because y ≠ 2
5. 5 · Check that x is in the domain. If (3y + 1)/(y − 2) = 3, then 3y + 1 = 3y − 6, that is 1 = −6, which is impossible. So x ≠ 3.
6. 6 · Verify f(x) = y. 2x + 1 = 7y/(y − 2) and x − 3 = 7/(y − 2), so f(x) = 7y/7 = y.
7. 7 · Check and box the answer. f is one-one and onto, so f is a bijection from R − {3} to R − {2}.

Scoring tip: Do not stop after finding x. Showing that x ≠ 3 and that f(x) = y earns the last marks.