Domain Of The Function F(x) = √(3x + 2) + 1 Explained

In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. Understanding the domain of a function, which represents the set of all possible input values for which the function is defined, is crucial for comprehending its behavior and applications. This article delves into the process of determining the domain of the function f(x) = √(3x + 2) + 1, providing a step-by-step explanation and highlighting the key concepts involved.

Demystifying the Domain: A Foundation for Function Analysis

Before we embark on our exploration of the specific function at hand, let's first establish a solid understanding of what the domain of a function entails. In essence, the domain encompasses all real numbers that can be validly inputted into the function without resulting in any undefined or non-real outputs. Certain mathematical operations, such as taking the square root of a negative number or dividing by zero, lead to undefined results in the realm of real numbers. Therefore, when determining the domain of a function, we must carefully identify and exclude any input values that would lead to such operations.

Keywords such as domain, function, real numbers, and undefined are central to this concept. A clear grasp of these terms is essential for navigating the intricacies of function analysis. For instance, when encountering a square root function, we must ensure that the expression under the radical is non-negative, as the square root of a negative number is not a real number. Similarly, when dealing with a rational function (a function expressed as a fraction), we must exclude any input values that would make the denominator equal to zero, as division by zero is undefined.

Understanding the domain of a function is paramount for several reasons. Firstly, it allows us to accurately interpret the function's behavior and graph. By knowing the set of permissible input values, we can avoid plotting points that lie outside the function's defined range. Secondly, the domain plays a critical role in various mathematical operations, such as finding the inverse of a function or solving equations involving functions. Without a clear understanding of the domain, we may arrive at erroneous conclusions or miss crucial solutions. Finally, the concept of domain extends far beyond theoretical mathematics, finding practical applications in diverse fields such as physics, engineering, and economics. For example, in physics, the domain of a function might represent the range of valid physical quantities, such as time or distance. In economics, the domain could represent the set of possible prices or quantities in a market model.

Unraveling the Domain of f(x) = √(3x + 2) + 1: A Step-by-Step Approach

Now that we have established a firm understanding of the concept of domain, let's turn our attention to the specific function in question: f(x) = √(3x + 2) + 1. To determine its domain, we must identify any restrictions on the input values, x, that would lead to undefined or non-real outputs. In this case, the primary restriction arises from the square root operation. As we discussed earlier, the expression under the radical (the radicand) must be non-negative for the function to produce real-valued outputs.

Step 1: Identify the Radicand

The radicand in our function is the expression 3x + 2. This is the expression that sits inside the square root symbol. Our goal is to ensure that this expression is always greater than or equal to zero.

Step 2: Set up the Inequality

To ensure the radicand is non-negative, we set up the following inequality:

3x + 2 ≥ 0

This inequality expresses the condition that the expression 3x + 2 must be greater than or equal to zero for the function to be defined in the realm of real numbers.

Step 3: Solve the Inequality

Now, we need to solve this inequality for x. To do this, we follow the standard algebraic steps for solving inequalities:

1. Subtract 2 from both sides of the inequality:

   3x ≥ -2

2. Divide both sides of the inequality by 3:

   x ≥ -2/3

This final inequality, x ≥ -2/3, represents the solution to our problem. It tells us that the function f(x) = √(3x + 2) + 1 is defined for all real numbers x that are greater than or equal to -2/3.

Step 4: Express the Domain in Interval Notation

To express the domain in interval notation, we use the following notation:

[-2/3, ∞)

This interval notation indicates that the domain includes all real numbers from -2/3 (inclusive, as indicated by the square bracket) to positive infinity. The parenthesis next to infinity indicates that infinity is not a specific number and is not included in the domain.

Therefore, the domain of the function f(x) = √(3x + 2) + 1 is the set of all real numbers greater than or equal to -2/3. This means that we can input any value of x that is -2/3 or larger into the function, and it will produce a real number output. If we try to input a value of x that is less than -2/3, the expression under the square root will become negative, and the function will be undefined in the real number system.

Visualizing the Domain: A Graphical Perspective

A graphical representation can often provide valuable insights into the domain of a function. If we were to graph the function f(x) = √(3x + 2) + 1, we would observe that the graph starts at the point (-2/3, 1) and extends to the right. This visually confirms our earlier finding that the domain of the function is all real numbers greater than or equal to -2/3. The graph does not exist for x values less than -2/3, reinforcing the restriction imposed by the square root operation.

Common Pitfalls to Avoid: Ensuring Accuracy in Domain Determination

When determining the domain of a function, it is essential to be mindful of common pitfalls that can lead to incorrect results. One frequent mistake is overlooking the restrictions imposed by square roots and other radical expressions. Remember that the radicand must always be non-negative. Another common error is forgetting to exclude values that would make the denominator of a rational function equal to zero. It's crucial to meticulously examine the function and identify any operations that could lead to undefined results.

Additionally, it's important to pay close attention to the inequality signs when solving inequalities. A simple mistake in the direction of the inequality can drastically alter the domain. Always double-check your work and ensure that your solution accurately reflects the restrictions imposed by the function.

By avoiding these common pitfalls and following a systematic approach, you can confidently determine the domain of a wide range of functions.

Conclusion: Mastering Domain Determination for Function Mastery

In this comprehensive guide, we have explored the concept of the domain of a function, focusing on the specific example of f(x) = √(3x + 2) + 1. We have demonstrated a step-by-step approach to determining the domain, emphasizing the importance of identifying restrictions imposed by mathematical operations such as square roots. By understanding the domain of a function, we gain a deeper appreciation for its behavior and limitations, paving the way for more advanced mathematical analysis.

Mastering domain determination is a crucial step in the journey towards function mastery. By practicing these techniques and remaining mindful of potential pitfalls, you can confidently tackle a wide variety of functions and unlock their hidden mathematical secrets.

The answer to the question "What is the domain of the function f(x) = √(3x + 2) + 1?" is D. all real numbers greater than or equal to -2/3.