91Ó°ÊÓ

The domain of a function is the set of all possible input values, or pairs in the case of multivariable functions, for which the function is defined. In simple terms, it's the collection of values you can put into the function without breaking any mathematical rules.

The function given in the exercise is a multivariable function: \[ f(x, y) = \frac{1}{\ln(4-x^2-y^2)} \] To figure out where this function is defined, we focused on the denominator because dividing by zero is undefined in mathematics. Moreover, the natural log function, \( \ln \), is only defined for positive values. So, the expression inside the logarithm, \( 4-x^2-y^2 \), needs to be greater than zero. This is our main constraint.

Finding the domain implies first solving: \[ 4 - x^2 - y^2 > 0 \]This translates to checking which points \((x, y)\) make this inequality true. If the expression inside the logarithm remains positive, then the function has a defined domain. In our case, it means verifying which \((x, y)\) pairs lie within a certain region in the xy-plane.

Inequalities, particularly in two variables, are crucial in defining regions in a plane. They often describe areas that satisfy a particular condition. In the given exercise, we are dealing with a quadratic inequality: \[ x^2 + y^2 < 4 \] This inequality tells us that we are looking for all \((x, y)\) points where the sum of the squares of \(x\) and \(y\) is less than 4.

Visualizing this inequality helps tremendously. Imagine a circle, centered at the origin \((0,0)\), with a radius of 2. The inequality \( x^2 + y^2 < 4 \) describes all the points inside this circle. This is because the square of the distance to the origin needs to be less than the square of the radius for points inside the circle.

Here are the key takeaways about inequalities in two variables:

• They can define regions on a plane.
• When an inequality includes less than (<), it describes the interior of the shape, not including the boundary.
• When it includes less than or equal to (≤), it also includes the boundary.

This specific inequality does not include the boundary, hence the domain excludes the circle's edge.

Sketching graphs helps in visualizing the solution of inequalities as well as understanding the domain of functions. In this exercise, sketching involves plotting a circle based on the inequality \( x^2 + y^2 < 4 \).

Here’s how you sketch it:

• First, identify the center of the circle. In our case, it is at coordinate origin \((0,0)\).
• Second, determine the radius of the circle, which is 2, from the inequality \( x^2 + y^2 < 4 \).
• Third, draw a circle centered at the origin with the specified radius.
• Lastly, shade the area inside the circle to indicate the domain. The circle's boundary is not included because of the '<' sign in the inequality.

Sketching is a visual method that provides insight into the nature of functions and their domains. It allows you to see, at a glance, how different values, or pairs of values, can fit into or affect the overall function. Visually interpreting these graphs can also aid in identifying which transformations might affect the domain, confirming solutions, or spotting mistakes.