Analyzing the Equation x2 y2 1: Even, Odd, or Neither?

When determining whether the function defined by the equation x2 y2 1 is even, odd, or neither, we first need to express y in terms of x. The equation x2 y2 1 describes a circle with a radius of 1 centered at the origin. Solving for y, we get:

y ±sqrt{1 - x2}

Even Function

For a function f(x) to be considered even, it must satisfy the condition:

f(-x) f(x) for all x

Odd Function

A function f(x) is considered odd if:

f(-x) -f(x) for all x

Analysis

Checking if it is an Even Function

For y sqrt{1 - x2}: f(-x) sqrt{1 - (-x)2} sqrt{1 - x2} f(x) For y -sqrt{1 - x2}: f(-x) -sqrt{1 - (-x)2} -sqrt{1 - x2} -f(x)

Conclusion

The function y sqrt{1 - x2} is even. The function y -sqrt{1 - x2} is odd.

Since the original equation x2 y2 1 can be derived from two functions, one of which is even and the other is odd, the equation does not fit strictly into the categories of even or odd functions. Therefore, we conclude that the relation x2 y2 1 is neither an even function nor an odd function.

Understanding Even and Odd Functions

An even function satisfies the condition f(x) f(-x). Examples of even functions include cos(x), and x2.

An odd function satisfies the condition f(x) -f(-x). Examples of odd functions include sin(x), and x3.

The term xn is an even function if n is even, and an odd function if n is odd. This is the basis from which these definitions arise.

Examining x2y 1

If we consider the equation x2y 1, it should be noted that y 1 - x2 is an even function. However, this is not the complete story, as this equation also determines three implicit functions:

y 1 - x2, an even function. x ±sqrt{1 - y}, which are neither odd nor even.

Is it a Function?

The equation x2 y2 1 is not a function in the strict sense. If we draw a vertical line through the circle, it will intersect the circle at more than one point, violating the definition of a function. Therefore, the question of whether it is odd or even does not apply.

However, the equation is symmetric both around the y-axis and the origin. Mathematically, this can be expressed as:

x - 02 y - 02 12 which simplifies to x2 y2 1.

In summary, the equation x2 y2 1 represents a circle and does not fit the categories of even or odd functions due to its symmetrical properties around both the y-axis and the origin.