Domain Of A Function Explained: Step-by-Step With Example

Hey guys! Today, we're diving into a crucial concept in mathematics: the domain of a function. It might sound a bit intimidating at first, but trust me, it's totally manageable once you grasp the basics. We'll break it down step by step, using a specific example to really solidify your understanding. So, let's get started and unlock the secrets of function domains!

What Exactly is the Domain of a Function?

In the simplest terms, the domain of a function is like its playground – it's the set of all possible input values (often called 'x' values) that you can feed into the function without causing any mathematical mayhem. Think of a function as a machine that takes an input, does some calculations, and spits out an output (often called 'y' or f(x)). The domain is all about what you're allowed to put into the machine. Some functions are pretty easygoing – you can plug in almost any number you can think of. But others are a bit more picky, having certain restrictions on what they can handle. These restrictions usually arise from two main culprits: division by zero and taking the square root (or any even root) of a negative number.

Let's delve deeper into these restrictions. Division by zero is a big no-no in mathematics. It's like trying to divide a pizza among zero people – it just doesn't make sense! So, any value of 'x' that would make the denominator of a fraction equal to zero is automatically excluded from the domain. This is because division by zero results in an undefined expression, which throws a wrench in our function's ability to produce a meaningful output. The second major restriction comes from even roots of negative numbers. You might remember that the square root of a negative number (like √-4) is not a real number; it's an imaginary number. Since we typically deal with real-valued functions (functions that produce real number outputs), we need to avoid any input values that would lead to taking an even root of a negative number. This means that the expression under the radical (the radicand) must be greater than or equal to zero. Understanding these two restrictions is paramount to accurately determine the domain of a function. We need to be vigilant in identifying any values of 'x' that would lead to either division by zero or taking an even root of a negative number, and then exclude them from the domain. By carefully considering these restrictions, we can ensure that our function operates smoothly and produces valid, real-number outputs.

Analyzing Our Example Function: f(x) = (x+6) / ((x-7)(x+5))

Now, let's put our knowledge into practice with a specific example. We have the function f(x) = (x+6) / ((x-7)(x+5)). This function is a rational function, meaning it's a fraction where both the numerator (the top part) and the denominator (the bottom part) are polynomials. The key to finding the domain of a rational function is to focus on the denominator. Remember, we can't divide by zero, so we need to figure out what values of 'x' would make the denominator equal to zero. In our case, the denominator is (x-7)(x+5). To find the values of 'x' that make this equal to zero, we set the entire expression equal to zero and solve for 'x': (x-7)(x+5) = 0. This equation is already factored for us, which is super helpful! To solve it, we use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero: x - 7 = 0 or x + 5 = 0. Solving the first equation, x - 7 = 0, we add 7 to both sides to get x = 7. Solving the second equation, x + 5 = 0, we subtract 5 from both sides to get x = -5. These are the two values of 'x' that make the denominator equal to zero. This means that if we plug in x = 7 or x = -5 into our function, we'll end up with division by zero, which is undefined. Therefore, these two values must be excluded from the domain.

Now that we've identified the problematic values, we can express the domain of the function. The domain consists of all real numbers except for 7 and -5. We can write this in a few different ways. One way is to use set notation: {x | x ∈ ℝ, x ≠ 7, x ≠ -5}. This reads as "the set of all x such that x is a real number and x is not equal to 7 and x is not equal to -5." Another way is to use interval notation. In interval notation, we represent the domain as a combination of intervals that include all the allowable values of 'x'. Since we're excluding 7 and -5, we'll have three intervals: (-∞, -5), (-5, 7), and (7, ∞). The parentheses indicate that the endpoints (-5 and 7) are not included in the intervals. The infinity symbols (∞ and -∞) represent positive and negative infinity, respectively, and are always enclosed in parentheses because we can never actually reach infinity. So, the domain in interval notation is (-∞, -5) ∪ (-5, 7) ∪ (7, ∞). The symbol "∪" means "union," which indicates that we're combining these three intervals together. Both set notation and interval notation are common ways to express the domain of a function, so it's good to be familiar with both. In conclusion, by carefully analyzing the denominator of our example function and identifying the values that would lead to division by zero, we were able to successfully determine the domain. This process highlights the importance of understanding the restrictions that can limit the input values of a function.

The Correct Answer and Why

Based on our analysis, we've determined that the domain of the function f(x) = (x+6) / ((x-7)(x+5)) is all real numbers except for -5 and 7. This corresponds to option C. Options A and B are incorrect because they either miss one of the excluded values or incorrectly identify them. Option A only excludes 7, while option B excludes 5 and -7, which is a sign error. It's crucial to carefully solve for the values that make the denominator zero to avoid these kinds of mistakes. Choosing the correct answer requires a solid understanding of the concept of domain and the ability to apply it to a specific function. By correctly identifying the values that make the denominator zero, we can accurately determine the domain and avoid common pitfalls. Always double-check your work and make sure you've considered all possible restrictions on the input values.

Key Takeaways and Further Exploration

Alright guys, let's recap the key things we've learned today about the domain of a function. First and foremost, remember that the domain is the set of all possible input values ('x' values) that a function can accept without resulting in undefined operations, like division by zero or taking the even root of a negative number. Identifying these restrictions is the cornerstone of finding the domain. For rational functions (fractions with polynomials), the primary focus is on the denominator. We need to find the values of 'x' that make the denominator equal to zero and exclude them from the domain. This often involves setting the denominator equal to zero and solving the resulting equation. For functions involving even roots (like square roots, fourth roots, etc.), we need to ensure that the radicand (the expression under the root) is greater than or equal to zero. This means setting the radicand greater than or equal to zero and solving the inequality. Once you've identified the values to exclude or the range of allowable values, you can express the domain using set notation or interval notation. Both are valid and widely used ways to represent the domain. To further solidify your understanding, try practicing with different types of functions. Explore functions with more complex denominators, functions with multiple restrictions, and functions involving radicals. You can also delve into the concept of range, which is the set of all possible output values ('y' values) of a function. Understanding both domain and range provides a complete picture of a function's behavior. There are many online resources and textbooks that offer additional examples and explanations. Don't hesitate to seek out these resources and continue practicing. The more you work with domains and ranges, the more comfortable and confident you'll become. So keep exploring, keep questioning, and keep pushing your mathematical boundaries! Remember, mastering the domain of a function is a fundamental step in your mathematical journey, opening doors to a deeper understanding of functions and their applications. So, keep up the great work, and you'll be tackling even more complex concepts in no time!