Finding Domains Of Functions Expressing In Interval Form
In mathematics, the domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's the set of all values that you can plug into a function and get a valid output. Understanding how to determine the domain of a function is crucial in various areas of mathematics, including calculus, algebra, and analysis. This article provides a comprehensive guide on how to find the domain of different types of functions, with a focus on expressing the answer in interval notation using exact values. Interval notation is a way of writing sets of real numbers using intervals, which are defined by their endpoints. For example, the interval (a, b) represents all real numbers between a and b, excluding a and b, while the interval [a, b] includes a and b. The concept of exact values refers to expressing numbers without approximation, such as using the square root symbol (√) instead of a decimal approximation. By mastering the techniques presented in this guide, you will be well-equipped to determine the domains of a wide range of functions.
Function 1: f(x) = 1 / (3x² + 12x + 9)
Identifying Potential Restrictions
To find the domain of the function f(x) = 1 / (3x² + 12x + 9), we need to identify any values of x that would make the function undefined. In this case, the function is a rational function, meaning it involves a fraction. Rational functions are undefined when the denominator is equal to zero, as division by zero is not allowed in mathematics. Therefore, our primary task is to find the values of x that make the denominator, 3x² + 12x + 9, equal to zero. This involves solving a quadratic equation, which is a fundamental concept in algebra. Quadratic equations often arise in various mathematical contexts, and mastering their solution is essential for understanding many mathematical concepts. In this specific case, finding the zeros of the denominator is crucial for determining the domain of the rational function. By identifying these values, we can exclude them from the set of all possible inputs, thereby defining the function's domain. This process highlights the importance of recognizing potential restrictions based on the function's structure and applying appropriate algebraic techniques to determine the values that need to be excluded.
Solving for Zeros of the Denominator
To find the zeros of the denominator, we set 3x² + 12x + 9 equal to zero and solve for x. The equation is: 3x² + 12x + 9 = 0. We can simplify this equation by dividing both sides by 3, which gives us: x² + 4x + 3 = 0. Now, we have a simplified quadratic equation that is easier to solve. There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is the most straightforward approach. We look for two numbers that multiply to 3 and add up to 4. These numbers are 1 and 3. Thus, we can factor the quadratic as: (x + 1)(x + 3) = 0. Setting each factor equal to zero gives us the solutions: x + 1 = 0 or x + 3 = 0. Solving these linear equations, we find: x = -1 and x = -3. These are the values of x that make the denominator equal to zero, and therefore, they must be excluded from the domain of the function. The process of solving for zeros is a fundamental skill in algebra and is essential for various applications, including finding the domain of rational functions and analyzing the behavior of polynomial functions.
Expressing the Domain in Interval Notation
Since the function is undefined at x = -1 and x = -3, we need to exclude these values from the domain. The domain of the function includes all real numbers except for -1 and -3. In interval notation, we represent this as a union of intervals. The domain consists of three intervals: all numbers less than -3, all numbers between -3 and -1, and all numbers greater than -1. In interval notation, this is written as: (-∞, -3) ∪ (-3, -1) ∪ (-1, ∞). This notation indicates that the domain includes all real numbers from negative infinity up to -3 (but not including -3), then all numbers between -3 and -1 (but not including -3 and -1), and finally, all numbers from -1 to positive infinity (but not including -1). The use of parentheses () indicates that the endpoints are not included in the interval, while square brackets [] would indicate that the endpoints are included. The symbol ∪ represents the union of sets, meaning we combine all the intervals into a single set. Interval notation is a concise and standard way of expressing the domain of a function, particularly when it involves multiple intervals or unbounded regions. Mastering this notation is essential for communicating mathematical ideas clearly and effectively.
Function 2: f(x) = √(3x² + 12x + 9)
Identifying Potential Restrictions
For the function f(x) = √(3x² + 12x + 9), the restriction comes from the square root. The square root of a negative number is not a real number, so we need to ensure that the expression inside the square root, 3x² + 12x + 9, is greater than or equal to zero. This means we need to solve the inequality 3x² + 12x + 9 ≥ 0. Inequalities involving square roots are common in mathematics, and understanding how to solve them is crucial for determining the domain of functions with square roots. The key idea is to ensure that the expression under the square root is non-negative. This requirement stems from the definition of the square root function in the realm of real numbers. When dealing with square roots, it's essential to identify this restriction and formulate the appropriate inequality. This step sets the stage for finding the values of x that satisfy the condition, thereby defining the domain of the function. Identifying potential restrictions is a critical step in finding the domain of any function, and it requires a careful consideration of the function's structure and the mathematical operations involved.
Solving the Inequality
To solve the inequality 3x² + 12x + 9 ≥ 0, we first simplify it by dividing both sides by 3, which gives us x² + 4x + 3 ≥ 0. This is a quadratic inequality, which we can solve by first finding the roots of the corresponding quadratic equation x² + 4x + 3 = 0. As we found earlier, the roots are x = -1 and x = -3. These roots divide the number line into three intervals: (-∞, -3), (-3, -1), and (-1, ∞). To determine the intervals where the inequality x² + 4x + 3 ≥ 0 is satisfied, we can test a value from each interval in the inequality. For the interval (-∞, -3), let's test x = -4. Plugging this into the inequality, we get (-4)² + 4(-4) + 3 = 16 - 16 + 3 = 3, which is greater than 0. So, the inequality is satisfied in this interval. For the interval (-3, -1), let's test x = -2. Plugging this into the inequality, we get (-2)² + 4(-2) + 3 = 4 - 8 + 3 = -1, which is less than 0. So, the inequality is not satisfied in this interval. For the interval (-1, ∞), let's test x = 0. Plugging this into the inequality, we get (0)² + 4(0) + 3 = 3, which is greater than 0. So, the inequality is satisfied in this interval. Since the inequality is non-strict (i.e., it includes ≥), we also include the roots x = -3 and x = -1 in the solution. Solving inequalities is a crucial skill in mathematics, particularly when dealing with functions involving radicals or other restrictions. The technique of finding critical points (in this case, the roots of the quadratic equation) and testing intervals is a standard approach for solving inequalities.
Expressing the Domain in Interval Notation
Based on our analysis, the inequality 3x² + 12x + 9 ≥ 0 is satisfied for x ≤ -3 and x ≥ -1. In interval notation, this is expressed as: (-∞, -3] ∪ [-1, ∞). This notation indicates that the domain includes all real numbers from negative infinity up to -3 (including -3), and all real numbers from -1 to positive infinity (including -1). The use of square brackets [] indicates that the endpoints are included in the interval, which is necessary because the inequality is non-strict. The symbol ∪ represents the union of sets, meaning we combine the two intervals into a single set. This final representation clearly and concisely defines the domain of the function f(x) = √(3x² + 12x + 9). Expressing the domain in interval notation is a standard practice in mathematics, as it provides a clear and unambiguous way to represent the set of all possible input values for a function. The use of appropriate symbols and conventions is essential for effective mathematical communication.
Finding the domain of a function is a fundamental skill in mathematics. By understanding the potential restrictions imposed by different types of functions, such as rational functions and square root functions, and by applying appropriate algebraic techniques, we can determine the set of all possible input values for which the function is defined. Expressing the domain in interval notation provides a clear and concise way to communicate the result. The examples discussed in this article illustrate the process of finding the domain for two common types of functions, and the techniques presented can be applied to a wide range of functions. Mastering these concepts is essential for success in various areas of mathematics, including calculus, algebra, and analysis. Understanding domains is not just a technical skill; it's a fundamental aspect of understanding the behavior and properties of functions. By carefully considering the restrictions and applying the appropriate techniques, you can confidently determine the domain of any function.
