Which of these is the domain of the function f(x) = 3x/(3 − x)?

Correct answer: c {x ∈ ℝ : x ≠ 3}

The denominator 3 − x is zero only at x = 3.

Why the other options are wrong

• a {x ∈ ℝ : x > 3}: x < 3 works too, for example f(0) = 0; only x = 3 is excluded.
• b {x ∈ ℝ : x < 3}: x > 3 works too, for example f(4) = −12.
• d {x ∈ ℝ : x ≠ 0}: A zero numerator is harmless, f(0) = 0; the trouble is the denominator.