Domain Of F(x) = 1/√(11x - 22): Interval Notation Guide

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Understanding the domain of a function is crucial in mathematics. The domain represents all possible input values (x-values) for which the function produces a valid output. When dealing with functions involving square roots and fractions, certain restrictions apply. In this article, we will delve into the process of finding the domain of the function f(x) = 1/√(11x - 22) and expressing it in interval notation. Let's embark on this mathematical journey together!

Understanding Domain Restrictions

When working with functions, it's important to identify any restrictions on the input values that would lead to undefined results. Two primary restrictions come into play when dealing with square roots and fractions:

  1. Square Roots: The expression inside a square root must be non-negative (greater than or equal to zero). This is because the square root of a negative number is not a real number.
  2. Fractions: The denominator of a fraction cannot be equal to zero. Division by zero is undefined in mathematics.

Considering these restrictions, let's analyze the function f(x) = 1/√(11x - 22) to determine its domain.

Analyzing the Function f(x) = 1/√(11x - 22)

Our function, f(x) = 1/√(11x - 22), combines both a square root and a fraction. This means we need to address both restrictions mentioned earlier.

  • Restriction 1: Square Root: The expression inside the square root, (11x - 22), must be greater than zero. We cannot include zero in this case because the square root is in the denominator, and we cannot divide by zero. Thus, we need to solve the inequality:

    11x - 22 > 0

  • Restriction 2: Fraction: The denominator, √(11x - 22), cannot be equal to zero. However, we've already addressed this in the square root restriction, as we're ensuring that (11x - 22) is strictly greater than zero.

Solving the Inequality

To find the domain, we need to solve the inequality 11x - 22 > 0. Let's walk through the steps:

  1. Add 22 to both sides:

    11x > 22

  2. Divide both sides by 11:

    x > 2

This inequality tells us that the values of x that satisfy the condition must be greater than 2.

Therefore, any x-value greater than 2 will result in a real number output for the function. This is a crucial step in determining the function's domain, as it sets the boundaries for valid input values. We've successfully isolated x and found the key condition for the function to be defined. Understanding this inequality is essential for expressing the domain in interval notation, which we'll cover in the next section. Remember, the goal is to ensure the expression inside the square root remains positive, avoiding any undefined results.

Expressing the Domain in Interval Notation

Now that we've determined that x > 2, we can express the domain of the function in interval notation. Interval notation is a concise way to represent a set of numbers using intervals.

  • The parenthesis '(' and ')' indicate that the endpoint is not included in the interval.
  • The bracket '[' and ']' indicate that the endpoint is included in the interval.
  • Infinity (∞) and negative infinity (-∞) are used to represent unbounded intervals.

In our case, x > 2 means that x can take any value greater than 2, but it cannot be equal to 2. Therefore, in interval notation, the domain of f(x) = 1/√(11x - 22) is:

(2, ∞)

This notation signifies that the domain includes all real numbers from 2 (exclusive) to infinity. Visualizing this on a number line can be helpful: imagine an open circle at 2 (indicating it's not included) and an arrow extending to the right, representing all numbers greater than 2. Understanding how to convert inequalities to interval notation is a fundamental skill in mathematics, especially when dealing with domains and ranges of functions. This notation provides a clear and concise way to express the set of all possible input values for which the function is defined.

Graphically Verifying the Domain

Visualizing the function's graph can provide a helpful confirmation of the domain we calculated. If you were to graph f(x) = 1/√(11x - 22), you would observe the following:

  • The graph does not exist for x ≤ 2. This aligns with our domain restriction that x must be strictly greater than 2.
  • The graph starts approaching the vertical line x = 2 but never touches it, indicating that 2 is not included in the domain.
  • The graph extends infinitely to the right, confirming that all values of x greater than 2 are part of the domain.

Graphing tools and software can be invaluable in verifying domain restrictions and understanding the behavior of functions. By visually inspecting the graph, you can reinforce your understanding of the algebraic calculations and ensure that the domain you've determined is consistent with the function's graphical representation. This graphical verification step adds another layer of confidence to your solution and helps solidify your grasp of the relationship between functions and their domains.

Common Mistakes to Avoid

When finding the domain of functions, there are some common pitfalls to watch out for. Here are a few to keep in mind:

  1. Forgetting the Square Root Restriction: Always remember that the expression inside a square root must be non-negative. Neglecting this can lead to an incorrect domain.
  2. Ignoring the Denominator: The denominator of a fraction cannot be zero. Make sure to identify any values of x that would make the denominator zero and exclude them from the domain.
  3. Incorrectly Solving Inequalities: Pay close attention to the rules of inequality manipulation. For example, multiplying or dividing both sides by a negative number reverses the inequality sign.
  4. Misinterpreting Interval Notation: Ensure you understand the difference between parentheses and brackets in interval notation. Parentheses indicate exclusion of the endpoint, while brackets indicate inclusion.

By being mindful of these common mistakes, you can improve your accuracy in determining the domains of various functions. Practice and careful attention to detail are key to mastering this concept.

Examples and Practice Problems

To further solidify your understanding of finding domains, let's look at some additional examples and practice problems.

Example 1:

Find the domain of g(x) = √(x - 5).

  • Solution: The expression inside the square root, (x - 5), must be greater than or equal to zero. x - 5 ≥ 0 x ≥ 5
  • Domain in interval notation: [5, ∞)

Example 2:

Find the domain of h(x) = 1/(x + 3).

  • Solution: The denominator, (x + 3), cannot be equal to zero. x + 3 ≠ 0 x ≠ -3
  • Domain in interval notation: (-∞, -3) ∪ (-3, ∞)

Practice Problems:

  1. Find the domain of f(x) = √(2x + 4).
  2. Find the domain of g(x) = 1/(x - 1).
  3. Find the domain of h(x) = √(x^2 - 9).

Working through these examples and practice problems will help you develop your skills in finding domains of various types of functions. Remember to carefully consider the restrictions imposed by square roots and fractions.

Conclusion

Finding the domain of a function is a fundamental skill in mathematics. By understanding the restrictions imposed by square roots and fractions, you can accurately determine the set of all possible input values for which the function is defined. In the case of f(x) = 1/√(11x - 22), we found that the domain is (2, ∞). Remember to always consider the restrictions, solve the relevant inequalities, and express the domain in interval notation. Keep practicing, and you'll master this important concept! For further learning, you can explore resources like Khan Academy's Domain and Range. This will give you a solid understanding of the topic. Remember, understanding the domain is crucial for understanding the behavior and properties of functions in mathematics.