Question

Challenge domains Find the domain of the following functions. Specify the domain mathematically, and then describe it in words or with a sketch. $$f(x, y, z)=\ln \left(z-x^{2}-y^{2}+2 x+3\right)$$

Short answer

Question: Determine the domain of the function $$f(x, y, z)=\ln(z-x^{2}-y^{2}+2x+3)$$. Answer: The domain of the function $$f(x, y, z)=\ln(z-x^{2}-y^{2}+2x+3)$$ is defined by the inequality $$z > (x-1)^2 + y^2 + 4$$, which represents an elliptical paraboloid opening in the positive z-direction with vertex (1,0,-4). All points in the three-dimensional space above and on the paraboloid satisfy the inequality, and thus belong to the domain.

Step-by-step solution

Rewrite inequality

First, we need to rewrite the inequality $$z-x^{2}-y^{2}+2x+3>0$$ in a more convenient form for further analysis. Start by moving the x and y squared terms to the right side to get the inequality: $$z > x^2 + y^2 - 2x + 3$$ Now, complete the squares for x and y to get a better understanding of the domain.

Completing the squares

We'll complete the squares for the x and y terms. Complete the square for the x part by adding and subtracting $$1$$: $$x^2-2x +1$$ Now, for the y part, we already have a perfect square, so we don't need to do anything. Now rewrite the inequality: $$z > (x-1)^2 + y^2 + 4$$

Describe the domain

The domain of the function consists of all triples (x, y, z) such that the inequality $$z > (x-1)^2 + y^2 + 4$$ is satisfied. In words, the domain is the set of all points (x, y, z) in which z is greater than the square of the (x-1) distance plus the square of the y distance plus 4. This represents an elliptical paraboloid opening in the positive z-direction with vertex (1,0,-4).

Sketch the domain

To sketch the domain, draw an elliptical paraboloid opening in the positive z-direction with vertex (1,0,-4). The x and y axes pass through the vertex, and the z-axis is perpendicular to the vertical axis of the paraboloid. The domain is the space above and on the paraboloid surface, since this is where the inequality is satisfied. To conclude, the domain of the given function $$f(x, y, z)=\ln(z-x^{2}-y^{2}+2x+3)$$ is defined by the inequality $$z > (x-1)^2 + y^2 + 4$$, which represents an elliptical paraboloid opening in the positive z-direction with vertex (1,0,-4). All points in the three-dimensional space above and on the paraboloid satisfy the inequality, and thus belong to the domain.

Key concepts

Domain of a Function

Understanding the domain of a function is essential in calculus, as it represents the set of all possible input values for which the function is defined. The domain specifies the permissible x, y, and z values that we can plug into the function without causing undefined behavior, such as taking the logarithm of a negative number or dividing by zero.

In the given exercise, we have a multivariable function involving a natural logarithm, denoted as \( f(x, y, z) \). It's important to note that the natural logarithm function, \( \ln(x) \), is only defined for positive values of x. Therefore, to find the domain, we look for all the values of x, y, and z that make the argument of the logarithm positive. In mathematical terms, this translates to solving the inequality \( z - x^2 - y^2 + 2x + 3 > 0 \). The resulting inequality gives us a condition that all points (x, y, z) must satisfy to be included in the domain of the function.

Completing the Square

Completing the square is a technique used to transform quadratic expressions into a perfect square trinomial plus a constant. This method is very useful for analyzing and solving quadratic equations and inequalities.

In our exercise, completing the square helps us to rewrite the inequality in a form that better reveals the shape of the domain. For the x-term \( x^2 - 2x \), we add and subtract the square of half the coefficient of x, which is 1, to get \( (x - 1)^2 \). This does not change the value of the expression because adding and subtracting the same amount (1 in this case) maintains equality. The y-term was already a perfect square \( y^2 \), so no alteration was needed. After completing the square, the inequality takes the form \( z > (x - 1)^2 + y^2 + 4 \), which is much easier to visualize and understand in terms of the domain.

Elliptical Paraboloid

An elliptical paraboloid is a 3-D surface that can be visualized as an infinite stack of ellipses where each slice parallel to the base is an ellipse, and each slice perpendicular to the base is a parabola.

After completing the square in the given exercise, we see that the domain of the function is represented by an elliptical paraboloid. Specifically, the inequality \( z > (x - 1)^2 + y^2 + 4 \) describes an elliptical paraboloid that opens upwards along the positive z-axis. The lowest point on this surface, or the vertex, is at the point (1, 0, -4), and the domain consists of all points lying on and above this paraboloidal surface. To sketch this, we would start by plotting the vertex and then drawing the paraboloidal shape above it, ensuring it opens in the correct direction—in this case, towards increasing z-values. This visual aids in grasping the three-dimensional aspect of the domain for the given multivariable function.