Asked by j
inverse of f(x)= x^2 +3
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GPT-5 mini
AI
f(x) = x^2 + 3 is not one-to-one on all real x (x and −x give the same value), so it has no global inverse unless you restrict the domain.
Solve for the inverse symbolically:
1. Write y = x^2 + 3.
2. Swap x and y: x = y^2 + 3.
3. Solve for y: y = ±√(x − 3).
Thus the inverse relation is y = ±√(x − 3) with domain x ≥ 3. To get a function (single-valued inverse) choose a branch:
- If you restrict the original f to x ≥ 0 (domain [0, ∞)), then f: [0,∞) → [3,∞) is one-to-one and
f⁻¹(x) = √(x − 3), for x ≥ 3.
- If you restrict f to x ≤ 0 (domain (−∞, 0]), then
f⁻¹(x) = −√(x − 3), for x ≥ 3.
Pick the branch that matches the domain restriction you intend.
Solve for the inverse symbolically:
1. Write y = x^2 + 3.
2. Swap x and y: x = y^2 + 3.
3. Solve for y: y = ±√(x − 3).
Thus the inverse relation is y = ±√(x − 3) with domain x ≥ 3. To get a function (single-valued inverse) choose a branch:
- If you restrict the original f to x ≥ 0 (domain [0, ∞)), then f: [0,∞) → [3,∞) is one-to-one and
f⁻¹(x) = √(x − 3), for x ≥ 3.
- If you restrict f to x ≤ 0 (domain (−∞, 0]), then
f⁻¹(x) = −√(x − 3), for x ≥ 3.
Pick the branch that matches the domain restriction you intend.
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