Determining The Domain Of Functions F(x) And G(x) A Comprehensive Guide
This article delves into the crucial concept of a function's domain, focusing on two specific functions, f(x) = (x + 7) / (x² - 49) and g(x) = x / (x² + 81). We will explore how to determine the domain for each function, expressing our answers in interval notation. Understanding the domain of a function is fundamental in mathematics as it defines the set of all possible input values (x-values) for which the function produces a valid output.
Determining the Domain of f(x) = (x + 7) / (x² - 49)
To effectively determine the domain of the function f(x) = (x + 7) / (x² - 49), we must identify any values of x that would make the function undefined. In this particular case, the function is a rational function, meaning it is expressed as a fraction where both the numerator and denominator are polynomials. The primary concern with rational functions is the denominator. A rational function becomes undefined when the denominator equals zero, as division by zero is not a permissible operation in mathematics.
Identifying Potential Issues
Therefore, our first step is to pinpoint the values of x that cause the denominator, x² - 49, to equal zero. We can accomplish this by setting the denominator equal to zero and solving for x:
x² - 49 = 0
This equation is a difference of squares, which can be factored as follows:
(x - 7)(x + 7) = 0
Setting each factor equal to zero gives us the solutions:
x - 7 = 0 => x = 7
x + 7 = 0 => x = -7
These solutions, x = 7 and x = -7, are the values that make the denominator zero and thus must be excluded from the domain of f(x). In simpler terms, the function f(x) cannot accept 7 or -7 as input values because these values would lead to division by zero, which is undefined.
Expressing the Domain in Interval Notation
Now that we have identified the values that are not in the domain, we can express the domain using interval notation. Interval notation is a concise way to represent a set of real numbers using intervals. It utilizes parentheses and brackets to indicate whether the endpoints are included or excluded from the set.
The domain of f(x) includes all real numbers except for 7 and -7. Therefore, we can represent the domain as a union of three intervals:
(-∞, -7) ∪ (-7, 7) ∪ (7, ∞)
This notation signifies that the domain includes all numbers from negative infinity up to -7 (but not including -7), all numbers between -7 and 7 (not including -7 and 7), and all numbers from 7 to positive infinity (not including 7). This effectively excludes -7 and 7 from the domain while including all other real numbers.
In summary, the domain of f(x) is (-∞, -7) ∪ (-7, 7) ∪ (7, ∞). This meticulous process of identifying and excluding problematic values ensures that we are working with a well-defined function across its entire domain.
Determining the Domain of g(x) = x / (x² + 81)
Let's shift our attention to the function g(x) = x / (x² + 81). Similar to our approach with f(x), we need to determine if there are any values of x that would result in an undefined output. Again, this is a rational function, so our primary focus remains on the denominator.
Analyzing the Denominator
We need to identify if there are any real values of x that will make the denominator, x² + 81, equal to zero. We set the denominator equal to zero and attempt to solve for x:
x² + 81 = 0
Subtracting 81 from both sides, we get:
x² = -81
Here's where we encounter a crucial difference compared to the previous function. We are now faced with the task of finding a real number that, when squared, results in a negative number (-81).
The Significance of a Sum of Squares
In the realm of real numbers, squaring any number, whether positive or negative, will always yield a non-negative result. For instance, 5² = 25 and (-5)² = 25. Therefore, there is no real number x that, when squared, will equal -81. This fundamental property is critical to understanding the domain of g(x).
The expression x² + 81 represents a sum of squares. Sums of squares, in contrast to differences of squares, do not factor in the real number system. This means we cannot find two real factors that multiply to give us x² + 81.
Concluding the Domain
Since there is no real value of x that makes the denominator x² + 81 equal to zero, there are no restrictions on the domain of g(x). The function is defined for all real numbers.
Expressing the Domain in Interval Notation
The domain of g(x) includes all real numbers, which can be expressed in interval notation as:
(-∞, ∞)
This notation signifies that the domain extends from negative infinity to positive infinity, encompassing all real numbers.
Therefore, the domain of g(x) is (-∞, ∞). This demonstrates that not all rational functions have restricted domains; the specific structure of the denominator plays a crucial role in determining the domain.
Key Takeaways for Determining Domains
Understanding how to determine the domain of a function is a critical skill in mathematics. Here's a summary of the key concepts we've explored:
- Rational Functions and Denominators: For rational functions (functions expressed as a fraction), the primary concern is the denominator. The function is undefined where the denominator equals zero.
- Solving for Problematic Values: Set the denominator equal to zero and solve for x to identify values that must be excluded from the domain.
- Difference of Squares: Expressions like x² - a² (where a is a constant) can be factored as (x - a)(x + a).
- Sum of Squares: Expressions like x² + a² do not factor in the real number system and are always positive for real values of x.
- Interval Notation: Use interval notation to express the domain, using parentheses for values excluded from the domain and brackets for values included.
- All Real Numbers: If there are no restrictions, the domain is all real numbers, represented as (-∞, ∞).
By applying these principles, you can confidently determine the domain of a wide range of functions, ensuring a solid foundation for further mathematical exploration.
Practice Problems
To solidify your understanding of domains, try determining the domains of the following functions:
- h(x) = (x - 2) / (x² - 9)
- k(x) = 1 / (x² + 4)
- m(x) = √(x - 3) (Hint: Consider values that make the expression under the square root non-negative)
Working through these problems will reinforce the concepts discussed and enhance your problem-solving skills.
By mastering the techniques for determining the domain of functions, you unlock a deeper understanding of mathematical relationships and pave the way for more advanced concepts in algebra, calculus, and beyond. The domain is just the beginning – it's the foundation upon which we build our understanding of functions and their behavior.
