Domain Of F(x) = √(4x + 1) - 4 A Comprehensive Solution
Determining the domain of a function is a fundamental concept in mathematics, particularly in the study of functions and their behavior. The domain represents the set of all possible input values (often denoted as 'x') for which the function produces a valid output. In simpler terms, it's the range of 'x' values that you can plug into the function without encountering any mathematical errors or undefined results. For the function f(x) = √(4x + 1) - 4, we need to carefully consider the restrictions imposed by the square root. The expression inside the square root, (4x + 1), must be greater than or equal to zero, as the square root of a negative number is not defined within the realm of real numbers. This constraint forms the basis for finding the domain of the function. To find the domain, we solve the inequality 4x + 1 ≥ 0. Subtracting 1 from both sides gives 4x ≥ -1. Then, dividing both sides by 4 yields x ≥ -1/4. This inequality tells us that the domain of the function consists of all real numbers x that are greater than or equal to -1/4. We can express this domain in interval notation as [-1/4, ∞). This interval includes -1/4 because the square root of zero is zero, a valid result. The infinity symbol (∞) indicates that the domain extends indefinitely in the positive direction. Understanding the domain is crucial because it tells us where the function is well-behaved and where it is not. In this case, the function f(x) = √(4x + 1) - 4 is defined for all x values greater than or equal to -1/4. Any x value less than -1/4 would result in taking the square root of a negative number, which is not a real number. Therefore, the domain of the function is [-1/4, ∞), which represents all real numbers greater than or equal to -1/4. This is a key aspect of understanding the function's behavior and its graphical representation, as it tells us the range of x-values over which the function exists.
Understanding Domain Restrictions
When delving into the realm of functions, it's crucial to grasp the concept of domain restrictions. These restrictions are essentially limitations on the input values (x-values) that a function can accept while still producing valid, real-number outputs. Certain mathematical operations are notorious for imposing such restrictions, and understanding these limitations is paramount for accurately analyzing functions. One of the most common culprits behind domain restrictions is the square root function. The fundamental issue with square roots is that they cannot handle negative numbers within the system of real numbers. Attempting to take the square root of a negative number results in an imaginary number, which falls outside the scope of real-valued functions. Therefore, any expression residing under a square root sign must be greater than or equal to zero to ensure a valid output. Another significant source of domain restrictions is the presence of fractions with variables in the denominator. Division by zero is an undefined operation in mathematics, meaning that a function becomes undefined at any x-value that causes the denominator of a fraction to equal zero. To determine these points of exclusion, one must identify the values of x that make the denominator zero and subsequently exclude them from the function's domain. Logarithmic functions also introduce domain constraints. Logarithms are only defined for positive arguments, implying that the expression inside the logarithm must be strictly greater than zero. This restriction arises from the very nature of logarithms as the inverse operation of exponentiation. Exponential functions, on the other hand, generally have fewer domain restrictions. However, specific exponential functions, such as those involving fractional or negative exponents, might impose certain limitations depending on the base. Trigonometric functions, including sine, cosine, tangent, and their reciprocals, also have their own unique domain restrictions. For instance, the tangent function (tan x) becomes undefined at angles where the cosine function (cos x) equals zero, leading to vertical asymptotes on the graph of the tangent function. Recognizing and addressing these domain restrictions is essential for accurately interpreting and manipulating functions. Failing to account for these limitations can lead to incorrect results and misunderstandings about the function's behavior. By carefully examining the mathematical operations involved in a function, one can effectively identify and navigate potential domain restrictions, ensuring a thorough and accurate analysis.
Step-by-Step Solution for f(x) = √(4x + 1) - 4
To find the domain of the function f(x) = √(4x + 1) - 4, we must follow a systematic, step-by-step approach. This ensures that we correctly identify any restrictions on the input values (x-values) that would lead to undefined or non-real outputs. The primary restriction in this function stems from the square root. As previously mentioned, the expression inside a square root must be greater than or equal to zero to yield a real number result. Therefore, the first critical step is to set the expression under the square root, (4x + 1), greater than or equal to zero. This yields the inequality 4x + 1 ≥ 0. This inequality captures the fundamental constraint on the domain of the function. Our next step involves isolating x to determine the range of x-values that satisfy this inequality. We begin by subtracting 1 from both sides of the inequality: 4x + 1 - 1 ≥ 0 - 1, which simplifies to 4x ≥ -1. This step maintains the balance of the inequality while moving us closer to isolating x. To fully isolate x, we divide both sides of the inequality by 4: (4x) / 4 ≥ (-1) / 4, which simplifies to x ≥ -1/4. This resulting inequality, x ≥ -1/4, precisely defines the domain of the function. It states that the function f(x) is defined for all x-values that are greater than or equal to -1/4. In other words, any x-value less than -1/4 would cause the expression inside the square root to become negative, leading to a non-real result. To express this domain in interval notation, we use the notation [-1/4, ∞). The square bracket on the left indicates that -1/4 is included in the domain (since the square root of zero is a valid result), and the infinity symbol (∞) on the right signifies that the domain extends indefinitely in the positive direction. This interval notation provides a concise and standardized way to represent the domain of the function. In summary, the domain of f(x) = √(4x + 1) - 4 is [-1/4, ∞), meaning that the function is defined for all real numbers greater than or equal to -1/4. This domain is crucial for understanding the behavior and graph of the function, as it tells us the range of x-values over which the function exists.
Expressing the Domain in Interval Notation
Interval notation is a standardized and concise method for representing sets of real numbers, particularly intervals. It provides a clear and unambiguous way to describe the domain and range of functions, making it an essential tool in mathematical analysis. Understanding interval notation is crucial for effectively communicating mathematical concepts and solutions. The notation uses brackets and parentheses to indicate whether the endpoints of an interval are included or excluded. Square brackets [ ] signify that the endpoint is included in the interval, while parentheses ( ) indicate that the endpoint is excluded. This distinction is particularly important when dealing with inequalities and domain restrictions. For instance, the interval [a, b] represents all real numbers between a and b, inclusive, meaning that both a and b are part of the interval. In contrast, the interval (a, b) represents all real numbers between a and b, exclusive, meaning that a and b are not part of the interval. We can also have mixed intervals, such as [a, b), which includes a but excludes b, or (a, b], which excludes a but includes b. When expressing unbounded intervals, those that extend to infinity, we use the infinity symbol (∞) or negative infinity symbol (-∞). Since infinity is not a number, it cannot be included in an interval, so we always use parentheses with infinity symbols. For example, the interval [a, ∞) represents all real numbers greater than or equal to a, extending indefinitely in the positive direction. Similarly, the interval (-∞, b) represents all real numbers less than b, extending indefinitely in the negative direction. The interval (-∞, ∞) represents the set of all real numbers. To express the union of multiple intervals, we use the union symbol (∪). For example, if a function's domain consists of two separate intervals, such as (-∞, 0) and [2, ∞), we would write the domain as (-∞, 0) ∪ [2, ∞). This indicates that the function is defined for all real numbers less than 0 and for all real numbers greater than or equal to 2. Applying interval notation to the function f(x) = √(4x + 1) - 4, we found that the domain is x ≥ -1/4. In interval notation, this is expressed as [-1/4, ∞). The square bracket on -1/4 indicates that -1/4 is included in the domain, and the parenthesis on ∞ indicates that the interval extends indefinitely in the positive direction. This interval notation succinctly captures the domain of the function, providing a clear and unambiguous representation of the allowable input values.
Why Option A is the Correct Answer
After our comprehensive analysis, we've determined that the domain of the function f(x) = √(4x + 1) - 4 is indeed [-1/4, ∞). This conclusion stems from the fundamental restriction imposed by the square root function: the expression inside the square root must be greater than or equal to zero. By setting 4x + 1 ≥ 0 and solving for x, we arrived at the inequality x ≥ -1/4. This inequality directly translates to the interval [-1/4, ∞), which signifies all real numbers greater than or equal to -1/4. Option A, [-1/4, ∞), precisely matches this derived domain, making it the correct answer. It's crucial to understand why the other options are incorrect to fully grasp the solution. Option B, (-∞, -1/4], represents all real numbers less than or equal to -1/4. This is incorrect because plugging any x-value less than -1/4 into the function would result in taking the square root of a negative number, which is not a real number. For instance, if we tried x = -1/2, we would have √(4(-1/2) + 1) = √(-2 + 1) = √(-1), which is not a real number. Option C, (-∞, 1/4], represents all real numbers less than or equal to 1/4. While this interval includes -1/4, it also includes values less than -1/4, which we've established are not in the domain. Thus, option C is incorrect because it contains values that lead to non-real outputs. Option D, [1/4, ∞), represents all real numbers greater than or equal to 1/4. This option is incorrect because it excludes the values between -1/4 and 1/4, which are part of the domain. For example, x = 0 is within the domain since √(4(0) + 1) = √1 = 1, but it is not included in the interval [1/4, ∞). The correctness of option A is further reinforced by visualizing the graph of the function. The graph of f(x) = √(4x + 1) - 4 starts at the point (-1/4, -4) and extends to the right, indicating that the function is defined for all x-values greater than or equal to -1/4. There is no portion of the graph to the left of x = -1/4, confirming that values less than -1/4 are not in the domain. In conclusion, option A, [-1/4, ∞), is the correct answer because it accurately represents the set of all real numbers for which the function f(x) = √(4x + 1) - 4 produces a valid, real-number output. This determination is based on the fundamental principle that the expression inside a square root must be non-negative.
In summary, the domain of the function f(x) = √(4x + 1) - 4 is [-1/4, ∞), as determined by the restriction imposed by the square root. This means that the function is defined for all real numbers greater than or equal to -1/4. Option A is the correct answer.
