Domain and range

Answer

The domain is the set of all valid input values (x-values) for which the function is defined. Example: f(x) = √x has domain x ≥ 0 because you cannot take the square root of a negative number.

Answer

1. Division by zero — values of x that make the denominator = 0 must be excluded. Example: f(x) = 1/(x − 3) → x ≠ 3. 2. Square root of a negative — the expression inside √ must be ≥ 0. Example: f(x) = √(x + 4) → x ≥ −4.

Answer

The expression inside √ must be ≥ 0: x − 5 ≥ 0 → x ≥ 5. Domain: x ≥ 5 (or [5, ∞) in interval notation). At x = 5: f(5) = √0 = 0 ✓. At x = 4: f(4) = √(−1) — undefined ✗.

Answer

The denominator is x² − 9 = (x − 3)(x + 3). This equals zero when x = 3 or x = −3. The domain excludes x = 3 and x = −3, not x = 9. Always set the denominator equal to 0 and solve — do not guess.

Answer

The range is the set of all possible output values (y-values) that the function can produce. Example: f(x) = x² has range y ≥ 0 because squaring any real number gives a non-negative result.

Answer

Squaring any real number always gives a non-negative result: (−3)² = 9, 0² = 0. The output can never be negative. So no matter what x you input, f(x) ≥ 0. The minimum value is 0 (at x = 0); the function grows without limit as x → ±∞.

Answer

Since x² ≥ 0, we have x² + 3 ≥ 3. Range: g(x) ≥ 3 (or [3, ∞)). The minimum value is 3, reached at x = 0: g(0) = 0 + 3 = 3.

Answer

The square root function only outputs non-negative values: √x ≥ 0 for all x ≥ 0. Correct range: f(x) ≥ 0 (or [0, ∞)). The function cannot produce negative outputs — √9 = 3, not ±3.

Answer

Look at the graph horizontally — the domain is the set of x-values covered by the graph. Find the leftmost and rightmost x-values. Filled circle (●) = endpoint included. Open circle (○) = endpoint not included.

Answer

Look at the graph vertically — the range is the set of y-values covered by the graph. Find the lowest and highest y-values reached by the graph. A filled dot means that y-value is included; an open dot means it is excluded.

Answer

Domain: −2 ≤ x ≤ 6. Range: −3 ≤ y ≤ 8 (or −3 ≤ f(x) ≤ 8). IB also accepts interval notation: domain [−2, 6], range [−3, 8].

Answer

Domain → x-axis (horizontal). Range → y-axis (vertical). Memory trick: "D for domain, D for direction left-right (x-axis)." Domain = span of x-values; range = span of y-values.

Answer

A restricted domain limits the valid inputs to a practical range — not all mathematical values make sense. Examples: • Time t: must be t ≥ 0 (time cannot be negative). • Number of items n: must be a positive integer (you cannot buy half an item). • Distance d: must be d ≥ 0.

Answer

Domain: 0 ≤ t ≤ 15. t ≥ 0: time cannot be negative. t ≤ 15: V(15) = 1200 − 80(15) = 0 — the pool is empty; the model stops being valid.

Answer

No — x = 11 is outside the domain [2, 10]. The function is not defined for x = 11; the output is meaningless in this context. Always check inputs are within the stated domain before calculating.

Answer

n must be a non-negative integer (you cannot sell −3.7 units). A more appropriate domain is n ∈ {0, 1, 2, 3, ...} or n ≥ 0 with n ∈ ℤ. IB context questions often award a mark for recognising this restriction.