Understanding the Domain of a Function: Exploring (x^2 1)

When dealing with functions, one of the critical concepts to understand is the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this article, we will explore the domain of the function (f(x) frac{1}{x^2 1}), which plays a significant role in clarifying our understanding of function domains in the realm of both real and complex numbers.

Definition and General Principles of Domain

The domain of a function is the set of all values of (x) for which the function (f(x)) is defined. This involves identifying any restrictions on the input values that would render the function undefined. Common restrictions include:

The denominator of a rational function cannot be zero. For logarithmic functions, the argument must be positive. For square root functions, the argument must be non-negative.

Let's delve deeper into the function (f(x) frac{1}{x^2 1}) and identify its domain.

The Role of the Denominator in Determining the Domain

For the function (f(x) frac{1}{x^2 1}), the denominator (x^2 1) is a key factor in determining the domain. This expression is a quadratic polynomial that is always positive for all real numbers (x) because:

(x^2 geq 0) for all (x in mathbb{R}). (1 > 0). (x^2 1 > 1).

Therefore, the denominator (x^2 1) can never be zero for any real number (x).

Exclusion of Complex Numbers

While the function (f(x) frac{1}{x^2 1}) is defined for all real numbers, we must also consider the domain in the context of complex numbers. In the complex plane, (x) can take any value, and the expression (x^2 1) can be zero. However, the solutions to (x^2 1 0) are:

(x i) (x -i)

These are the imaginary units in the complex plane. For any complex number (x) except (i) and (-i), the expression (x^2 1) is non-zero. Therefore, the domain of (f(x)) is the set of all complex numbers excluding (i) and (-i).

Conclusion

In summary, the domain of the function (f(x) frac{1}{x^2 1}) is:

All real numbers (mathbb{R}). All complex numbers except (i) and (-i).

This function is well-behaved in both the real and complex domains, providing a valuable example for understanding function domains across different number systems.

References

[1] Wikipedia - Function (mathematics)

[2] MathWorld - Complex Number