Domain Of The Function Y = √(x) + 4 Explained

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In mathematics, the domain of a function is a fundamental concept. It defines the set of all possible input values (often denoted as x) for which the function will produce a valid output. In simpler terms, it's the range of x-values you can plug into a function without causing mathematical errors, such as dividing by zero or taking the square root of a negative number. Determining the domain is crucial for understanding the behavior and limitations of a function. For the function presented, y = √x + 4, we will explore the domain, which involves understanding the restrictions imposed by the square root operation.

The function y = √x + 4 is a simple yet illustrative example of how mathematical operations can restrict the domain. The square root function, √x, is defined only for non-negative real numbers. This is because the square root of a negative number is not a real number; it results in an imaginary number. Therefore, the expression under the square root, in this case, x, must be greater than or equal to zero. The addition of 4 outside the square root does not affect the domain, as it only shifts the function vertically and does not introduce any new restrictions on the possible values of x. Thus, the domain of y = √x + 4 is primarily determined by the requirement that x must be non-negative. To determine the domain of the given function, we need to identify any restrictions on the input variable x. The key restriction here comes from the square root function. The square root of a negative number is not defined within the set of real numbers. Therefore, the expression inside the square root, which is x, must be greater than or equal to zero. This condition ensures that we are only taking the square root of non-negative numbers, resulting in real number outputs for y. The '+ 4' part of the function only shifts the graph vertically and does not impact the domain, which is determined solely by the square root term. This is a crucial aspect to grasp when working with functions that involve radicals or other operations with inherent domain restrictions.

To further illustrate this point, let's consider a few examples. If x is a positive number, such as 9, then y = √9 + 4 = 3 + 4 = 7, which is a real number. If x is 0, then y = √0 + 4 = 0 + 4 = 4, which is also a real number. However, if x were a negative number, such as -1, then y = √(-1) + 4. The square root of -1 is an imaginary number (denoted as i), and adding 4 to it would still result in a complex number, not a real number. This violates the condition for the function to produce a real-valued output. Thus, the domain must exclude all negative numbers. This principle applies to any function involving a square root; the expression inside the square root must always be greater than or equal to zero to ensure the function yields real outputs. The domain of y = √x + 4 is the set of all non-negative real numbers, often written in interval notation as [0, ∞). Understanding this limitation is essential for graphing the function, solving equations involving it, and applying it in real-world contexts.

Step-by-Step Solution

  1. Identify the restriction: The square root function, denoted as √x, is defined only for non-negative values of x. This means that the expression inside the square root must be greater than or equal to zero.
  2. Set up the inequality: In the function y = √x + 4, the expression inside the square root is simply x. Therefore, we must have x ≥ 0.
  3. Solve the inequality: The inequality x ≥ 0 is already solved, indicating that x can be any non-negative real number.
  4. Express the domain in interval notation: The domain consists of all real numbers greater than or equal to 0. In interval notation, this is written as [0, ∞).

Detailed Explanation of the Answer Choices

To fully grasp the concept, let's examine why the other answer choices are incorrect:

  • A. -∞: This option is incomplete. While it suggests negative infinity, it doesn't provide an upper bound or a condition for x. It fails to capture the non-negative restriction imposed by the square root function.
  • B. -4 ≤ x < ∞: This choice is incorrect because it includes negative values for x down to -4. For instance, if x = -1, then y = √(-1) + 4, which involves the square root of a negative number, resulting in an imaginary number, not a real number. This violates the requirement for the function to have real-valued outputs. The square root of a negative number is undefined in the set of real numbers, making this domain invalid for the given function.
  • C. 0 ≤ x < ∞: This is the correct answer. It accurately represents the domain of the function y = √x + 4. The inequality 0 ≤ x ensures that x is non-negative, which is a necessary condition for the square root function to produce real number outputs. The inclusion of 0 means that x can be zero, and the function y will still be defined (y = √0 + 4 = 4). The upper bound of infinity (∞) indicates that x can be any non-negative real number without restriction. Thus, this domain encompasses all permissible input values for x in the function. The graphical representation of y = √x + 4 would start at the point (0, 4) and extend infinitely to the right, confirming that x must be non-negative.
  • D. 4 ≤ x < ∞: This option is incorrect because it implies that x must be greater than or equal to 4. However, values of x between 0 and 4, such as 1, 2, and 3, are perfectly valid inputs for the function. For example, if x = 1, then y = √1 + 4 = 5, which is a real number. Thus, this domain is too restrictive and excludes valid input values for x. It overlooks the fundamental condition that x only needs to be non-negative, not necessarily greater than or equal to 4.

Additional Considerations

Understanding the domain of a function is not just a theoretical exercise; it has practical implications. For example, when graphing the function, you would only plot points for x-values within the domain. In real-world applications, the domain may represent physical constraints or limitations. For instance, if x represents time, it cannot be negative. Similarly, if x represents a physical quantity like length or mass, it must be non-negative. Thus, understanding the domain helps in interpreting the function's behavior and applying it appropriately in various contexts. Moreover, the domain is crucial for performing further mathematical operations on the function, such as finding its inverse or analyzing its continuity. An incorrect domain can lead to erroneous results and misinterpretations of the function's properties. Therefore, careful consideration of the domain is a fundamental step in any mathematical analysis involving functions.

Conclusion

The domain of the function y = √x + 4 is 0 ≤ x < ∞, which means x can be any non-negative real number. This restriction arises from the square root function, which is only defined for non-negative inputs. Understanding the domain is essential for accurately working with and interpreting functions in mathematics and its applications.