Question

g(x)=\sqrt{x+3}

Answer 1

Here are key properties of g(x) = √(x + 3) = (x + 3)^{1/2}:

- Domain: x + 3 ≥ 0 ⇒ x ≥ −3.
- Range: y ≥ 0.
- Intercepts: g(−3) = 0 (x-intercept); g(0) = √3 (y-intercept).
- Monotonicity: strictly increasing on its domain.
- Continuity/differentiability: continuous for x ≥ −3; differentiable for x > −3. Not differentiable at x = −3 (vertical tangent / infinite slope).
- Derivative: g'(x) = 1 / (2√(x + 3)) for x > −3.
- Second derivative: g''(x) = −1 / (4(x + 3)^{3/2}) for x > −3 (concave down).
- Integral: ∫√(x + 3) dx = (2/3)(x + 3)^{3/2} + C.
- Inverse function: g^{-1}(y) = y^2 − 3 with domain y ≥ 0.
- Graph: the basic √x curve shifted left by 3 units; starts at point (−3, 0) and increases slowly to +∞ as x → +∞.

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