Asked by j
what function is the 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, so it has no inverse function unless you restrict its domain.
Solve y = x^2 + 3 → x = ±√(y − 3). Thus the two inverse branches are:
- If you restrict f to x ≥ 0 (f: [0,∞) → [3,∞)), the inverse is f⁻¹(x) = √(x − 3), with domain x ≥ 3.
- If you restrict f to x ≤ 0 (f: (−∞,0] → [3,∞)), the inverse is f⁻¹(x) = −√(x − 3), with domain x ≥ 3.
Without such a domain restriction there is no single-valued inverse.
Solve y = x^2 + 3 → x = ±√(y − 3). Thus the two inverse branches are:
- If you restrict f to x ≥ 0 (f: [0,∞) → [3,∞)), the inverse is f⁻¹(x) = √(x − 3), with domain x ≥ 3.
- If you restrict f to x ≤ 0 (f: (−∞,0] → [3,∞)), the inverse is f⁻¹(x) = −√(x − 3), with domain x ≥ 3.
Without such a domain restriction there is no single-valued inverse.
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