Domain Of Logarithmic Function F(x) = Log((-x² + 1)^(1/2)) Explained

Hey guys! Today, we're diving deep into the fascinating world of logarithmic functions, specifically focusing on finding the domain of the function f(x) = log((-x² + 1)^(1/2)). This might seem a bit daunting at first, but trust me, we'll break it down step by step and make it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Domain of Logarithmic Functions

Before we jump into the specifics of our function, let's quickly recap what the domain of a function actually means. In simple terms, the domain is the set of all possible input values (x-values) for which the function will produce a valid output (y-value). For logarithmic functions, there's a crucial restriction we need to keep in mind: the argument of the logarithm (the expression inside the log) must always be strictly greater than zero.

Think of it this way: logarithms are the inverse of exponential functions. Exponential functions always produce positive outputs, so logarithms can only accept positive inputs. Trying to take the logarithm of zero or a negative number is like trying to divide by zero – it's simply undefined in the realm of real numbers. This is the key concept we need to remember when determining the domain of a logarithmic function.

Now, let's translate this concept into the context of our function, f(x) = log((-x² + 1)^(1/2)). The argument of our logarithm is (-x² + 1)^(1/2), which is the square root of (-x² + 1). This means that for f(x) to be defined, two conditions must be met simultaneously. Firstly, the expression inside the square root, (-x² + 1), must be greater than or equal to zero because we can only take the square root of non-negative numbers in the real number system. Secondly, the entire argument of the logarithm, (-x² + 1)^(1/2), must be strictly greater than zero, as we've already established for all logarithmic functions. These two conditions form the backbone of our domain calculation.

Therefore, when dealing with logarithmic functions, especially those involving radicals, we must meticulously consider both the restrictions imposed by the logarithm itself and any restrictions imposed by other functions within the argument, such as square roots or fractions. By carefully analyzing these restrictions, we can accurately determine the set of permissible input values, which constitutes the domain of the function. In the following sections, we will delve into the algebraic manipulations required to solve the inequalities arising from these restrictions and ultimately define the domain of our specific function, f(x) = log((-x² + 1)^(1/2)). Remember, understanding these fundamental principles is crucial for mastering not just this particular problem, but also a wide range of problems involving logarithmic functions and their domains.

Step-by-Step Solution: Finding the Domain

Alright, let's get our hands dirty and actually solve for the domain of f(x) = log((-x² + 1)^(1/2)). As we discussed, the argument of the logarithm, (-x² + 1)^(1/2), must be greater than zero. This gives us our first inequality:

(-x² + 1)^(1/2) > 0

To get rid of the square root, we can square both sides of the inequality. Squaring both sides of an inequality is generally safe as long as we know that both sides are non-negative, which is true in our case since a square root is always non-negative and we're requiring it to be strictly greater than zero. This gives us:

(-x² + 1) > 0

Now, let's rearrange this inequality to make it easier to work with. We can add x² to both sides and subtract 1 from both sides to get:

1 > x²

Or, equivalently:

x² < 1

This inequality tells us that x² must be less than 1. To solve this, we can take the square root of both sides, remembering that when we take the square root of both sides of an inequality involving squares, we need to consider both the positive and negative roots. This gives us:

-1 < x < 1

This inequality tells us that x must be between -1 and 1, but not including -1 and 1 themselves. Why not including -1 and 1? Because if x were equal to -1 or 1, then -x² + 1 would be equal to 0, and the square root of 0 is 0. But we need the square root to be strictly greater than 0, not equal to 0, because the logarithm of 0 is undefined. Therefore, x cannot be -1 or 1.

So, our solution is the open interval (-1, 1). This means that any value of x between -1 and 1 (excluding -1 and 1) will result in a valid output for our function f(x) = log((-x² + 1)^(1/2)). This is the domain of our function. We've successfully navigated the restrictions imposed by both the square root and the logarithm to arrive at our final answer. This step-by-step approach is crucial for tackling more complex problems involving domains of functions.

Expressing the Domain in Different Notations

Now that we've found the domain, it's important to be able to express it in different notations. This helps in communicating the solution clearly and understanding it in various contexts. We've already seen the domain expressed as an inequality: -1 < x < 1. This is a very intuitive way to represent the range of values that x can take.

Another common way to express the domain is using interval notation. In interval notation, we use parentheses and brackets to indicate whether the endpoints are included or excluded. Parentheses indicate that the endpoint is not included, while brackets indicate that the endpoint is included. Since our domain includes all values between -1 and 1, but not -1 and 1 themselves, we use parentheses. Therefore, the domain in interval notation is written as:

(-1, 1)

This notation is concise and widely used in mathematics. It clearly conveys that the domain consists of all real numbers between -1 and 1, excluding the endpoints. Understanding interval notation is crucial for working with sets and ranges of values in various mathematical contexts.

Finally, we can also express the domain using set-builder notation. Set-builder notation provides a more formal way to define a set based on a specific condition. In this notation, we write:

{x ∈ ℝ | -1 < x < 1}

This reads as "the set of all x belonging to the set of real numbers (ℝ) such that x is greater than -1 and less than 1." Set-builder notation is particularly useful when dealing with more complex sets and conditions. It provides a precise and unambiguous way to define the domain of a function.

So, we've successfully expressed the domain of f(x) = log((-x² + 1)^(1/2)) in three different notations: inequality notation, interval notation, and set-builder notation. Being comfortable with these different notations is essential for effective mathematical communication and problem-solving. Each notation offers a slightly different perspective on the domain, and understanding them all allows you to choose the most appropriate notation for a given situation.

Common Mistakes and How to Avoid Them

Finding the domain of logarithmic functions, especially those with radicals, can be tricky. Let's talk about some common pitfalls and how to avoid them. One frequent mistake is forgetting the fundamental restriction of logarithms: the argument must be strictly greater than zero. People sometimes mistakenly allow the argument to be greater than or equal to zero, which is incorrect. Remember, log(0) is undefined!

Another common error arises when dealing with square roots. While it's true that the expression inside a square root can be zero, the entire argument of the logarithm cannot be zero. So, in our example, (-x² + 1) can be equal to zero, but (-x² + 1)^(1/2) cannot be zero. This subtle distinction is crucial for accurately determining the domain.

A third mistake involves squaring both sides of an inequality. As we discussed earlier, squaring both sides is generally safe when both sides are non-negative. However, if one or both sides are negative, squaring can flip the inequality sign, leading to an incorrect solution. In our case, we were careful to ensure that both sides were non-negative before squaring, so this wasn't an issue. But it's a point to keep in mind for other problems.

Finally, a lack of attention to detail can also lead to errors. When solving inequalities, it's important to carefully track the signs and ensure that all steps are performed correctly. A simple arithmetic mistake can throw off the entire solution. It's always a good idea to double-check your work to minimize the risk of errors.

To avoid these mistakes, it's helpful to follow a systematic approach. First, identify all the restrictions imposed by the function, such as the argument of a logarithm being positive or the expression inside a square root being non-negative. Second, translate these restrictions into inequalities. Third, solve the inequalities carefully, paying attention to the rules of inequality manipulation. Fourth, express the domain in a clear and concise notation, such as interval notation or set-builder notation. By following these steps, you can significantly reduce the likelihood of making errors and confidently determine the domain of even complex logarithmic functions.

Conclusion: Mastering the Domain of Logarithmic Functions

So, there you have it! We've successfully navigated the intricacies of finding the domain of the logarithmic function f(x) = log((-x² + 1)^(1/2)). We've covered the fundamental principles of logarithmic functions, the restrictions they impose, and the algebraic techniques required to solve for the domain. We've also discussed common mistakes and how to avoid them, and we've explored different ways to express the domain in mathematical notation.

Understanding the domain of a function is a crucial skill in mathematics. It allows us to define the set of valid inputs for a function and ensures that we're working within the realm of meaningful results. Mastering this skill is essential for success in calculus, analysis, and other advanced mathematical topics. Logarithmic functions, with their unique restrictions, provide a great opportunity to practice and solidify our understanding of domains.

Remember, the key to finding the domain of a logarithmic function is to ensure that the argument of the logarithm is strictly greater than zero. When other functions, such as square roots, are involved, we need to consider their restrictions as well. By carefully analyzing these restrictions and solving the resulting inequalities, we can confidently determine the domain.

So, keep practicing, keep exploring, and keep pushing your mathematical boundaries! The world of functions and their domains is vast and fascinating, and there's always something new to discover. And remember, don't be afraid to ask questions and seek help when you need it. We're all in this together, learning and growing in our mathematical journey.