Example 1: evaluating a function

• If f(x) = 3x − 5, find f(2) and f(−1).
• Solution f(2) = 3 × 2 − 5 = 6 − 5 = 1. And f(−1) = 3 × (−1) − 5 = −3 − 5 = −8.

Example 2: domain of a square-root function

1. Find the domain of f(x) = √(x − 2).
2. We need x − 2 ≥ 0, so x ≥ 2. The domain is [2, ∞[.

Example 3: domain of a rational function

• Find the domain of f(x) = 1 / (x² − 9).
• Solution The denominator must not be zero, so x² − 9 ≠ 0. Factoring gives (x − 3)(x + 3) ≠ 0, so x ≠ 3 and x ≠ −3. The domain is all real numbers except −3 and 3.

Example 4: a function with two conditions

• Find the domain of f(x) = √(6 − 2x) / (x + 1).
• Solution First condition: 6 − 2x ≥ 0, so 2x ≤ 6 and x ≤ 3. Second condition: x + 1 ≠ 0, so x ≠ −1. The domain is ]−∞, 3] with the value −1 removed.

Example 5: finding x from a known output