G(x,y) = sqrt(1 + x^2 + y^2), please identify the surface and state the domain and range of the function.

The given function is:

\[
G(x, y) = \sqrt{1 + x^2 + y^2}
\]

### 1. Identifying the Surface:
This function represents a **paraboloid** (specifically, a circular paraboloid) in 3D space. A circular paraboloid is a surface where the vertical distance \( G(x, y) \) from the plane is a function of the sum of the squares of the horizontal distances \( x^2 \) and \( y^2 \).

Geometrically, the equation:

\[
z = \sqrt{1 + x^2 + y^2}
\]

describes a surface that looks like a bowl (a paraboloid), but it’s been vertically shifted by 1 unit, starting from \( z = 1 \) at the origin (when \( x = 0 \) and \( y = 0 \)).

### 2. Domain of \( G(x, y) \):
The domain is the set of all possible input values \( (x, y) \). Since the square root function is defined for non-negative values of its argument, and \( 1 + x^2 + y^2 \geq 1 \) for all real numbers \( x \) and \( y \), the domain is:

\[
\text{Domain}: \mathbb{R}^2 = \{(x, y) \in \mathbb{R}^2 \}
\]

This means the domain includes all real values for \( x \) and \( y \).

### 3. Range of \( G(x, y) \):
The range is the set of all possible output values \( G(x, y) \). The minimum value of \( 1 + x^2 + y^2 \) is 1, which occurs when \( x = 0 \) and \( y = 0 \). As \( x \) and \( y \) increase, \( G(x, y) \) increases without bound. Therefore, the range of \( G(x, y) \) is:

\[
\text{Range}: [1, \infty)
\]

### Summary:
- The surface represented by \( G(x, y) = \sqrt{1 + x^2 + y^2} \) is a **paraboloid**.
- **Domain**: All real values for \( x \) and \( y \), or \( \mathbb{R}^2 \).
- **Range**: \( [1, \infty) \), starting from 1 and increasing without bound.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the surface change if we added another term, like \( z = \sqrt{1 + x^2 + y^2 + z^2} \)?
2. What would happen to the domain and range if we used \( G(x, y) = \sqrt{x^2 + y^2} \) instead?
3. How do we find the tangent plane to the surface at a given point?
4. What is the general equation for a paraboloid, and how does it relate to ellipsoids?
5. How would the function behave if the square root were removed?

**Tip**: For surfaces like this, visualizing cross-sections along different axes can help in understanding their geometry.

Explore how to identify the surface represented by G(x, y) = sqrt(1 + x^2 + y^2), a circular paraboloid, and understand its domain and range. This problem involves key concepts from multivariable calculus, surface geometry, and paraboloid definitions. Ideal for college-level students studying calculus.

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