91Ó°ÊÓ

When we talk about functions of two variables, we mean a function that takes two input values--let's call them x and y--and produces an output value. In this specific exercise, our function is described as: \(f(x, y) = x^2 + y^2\). This means that for any given pair of x and y, we will get a single output value by squaring both x and y, and then adding the results.

Functions of two variables are common in mathematics and can be used to describe surfaces in a three-dimensional space. To visualize it, think of x and y as coordinates on the horizontal plane, and the result, f(x, y), as the height at that point. The function can represent various real-world situations such as the landscape of hills (where the height changes depending on the location).

In our function \(f(x, y) = x^2 + y^2\), the sum of squares plays a critical role. The term 'sum of squares' simply means adding the squares of given numbers. Here's how it works:

1. Take each variable (in our case, x and y).
2. Square them. So, x becomes \(x^2\) and y becomes \(y^2\).
3. Add the results together. So, \(x^2 + y^2\).

Squaring a number means multiplying the number by itself. For example, if x = 3, then \(x^2 = 3 \times 3 = 9\). Adding the squares of two numbers can give insights into various properties, like the distance from the origin in Cartesian coordinates. In our function, this concept is used in determining the value at (x, y) coordinates.

Now, let's talk about evaluating the function. We have our function as \(f(x, y) = x^2 + y^2\). Here's how to evaluate it step-by-step:

1. **Identify the values for x and y.** For example, let’s take x = 1 and y = 2.
2. **Square each of these values**: \(1^2 = 1\), and \(2^2 = 4\).
3. **Add the results**: \(1 + 4 = 5\).

So, the function evaluated at (1, 2) is 5, and we write this as \(f(1, 2) = 5\).

Evaluating functions of two variables helps us see how the function behaves at different points on a plane. For another example, if x = 0 and y = 3, then \(f(0, 3) = 0^2 + 3^2 = 0 + 9 = 9\). This process gives us the precise output for any pair of inputs we choose.