91Ó°ÊÓ

Function analysis is a crucial aspect of understanding how a function behaves in different conditions. For the given exercise, function analysis involves taking the function \( f(x, y) = x - y \) and determining which of the specified conditions it meets. This process can help reveal important properties of the function, such as symmetry or antisymmetry.
When analyzing a function, the goal is often to see how changes in the inputs affect the outputs.

• Identify the function: understanding what the function is doing, such as \( f(x, y) = x - y \), simplifies the exploratory process.
• Apply transformations: like substituting \((-x, -y)\) into the function to see how it responds to altered inputs.
• Determine satisfaction of conditions: this involves checking if transformations satisfy predefined equations, like \( f(-x, -y) = f(x, y) \) or \( f(-x, -y) = -f(x, y) \).

For our function, \( f(x, y) = x - y \), transforming it by substituting \((-x, -y)\) gives us \(-x + y\). By comparing this result to the conditions, we can determine which, if any, are satisfied. Function analysis helps reinforce understanding by providing a structured approach to checking for symmetry, which leads directly into concepts of odd and even functions.

In mathematics, a function is known as odd if it satisfies the condition \( f(-x) = -f(x) \). Odd functions have a symmetric property around the origin. Interestingly, they have a visual trait where their graph looks identical when rotated 180 degrees around the origin.
Odd functions can include both single-variable and multi-variable scenarios. For single-variable, think of functions like \( f(x) = x^3 \) or \( f(x) = -sin(x) \). Meanwhile, a multi-variable function, such as the one from our exercise \( f(x, y) = x - y \), can also be odd if it satisfies a similar multi-variable condition like \( f(-x, -y) = -f(x, y) \).
To confirm if our function \( f(x, y) = x - y \) is odd, we substitute \((-x, -y)\):

• Calculation gives \(-x + y\);
• Comparing results, \(-x + y = -(x - y)\), verifies the odd function condition.

The proof completes with confirming that the function meets the criterion for being classified as an odd function, making it symmetric about the origin in a multi-variable sense.

Even functions are another important class of functions in mathematics. These functions have a defining property: \( f(-x) = f(x) \).
This symmetry is about the y-axis for single-variable functions. Popular examples include \( f(x) = x^2 \) or \( f(x) = \cos(x) \). For a function to be even, substituting \(-x\) into \( f(x) \) returns the same original function value.
Despite their name, even functions are not about the number value but rather about their symmetrical property. For multi-variable functions, an even-like symmetry condition such as \( f(-x, -y) = f(x, y) \) can be established. For instance, a function \( f(x, y) = x^2 + y^2 \) would be considered even under this condition.

• Our function \( f(x, y) = x - y \) was tested against this even function condition and it failed;
• This is because \(-x + y\) resulting from substitution was not equal to the original equation \(x - y\).

Therefore, this particular function does not exhibit even symmetry.