Finding The Domain Of P(x) = √(x-1) + 2 A Step-by-Step Guide
In the realm of mathematics, particularly within the study of functions, the concept of a domain is fundamental. The domain of a function defines the set of all possible input values (often represented as 'x') for which the function produces a valid output. In simpler terms, it's the range of values that you can plug into a function without encountering any mathematical impossibilities, such as division by zero or taking the square root of a negative number. This article delves deep into understanding the domain of a specific function, $p(x) = \sqrt{x-1} + 2$, providing a step-by-step analysis and explanation to ensure clarity and comprehension. We will dissect the function, identify potential restrictions, and ultimately determine the correct domain, solidifying your understanding of this critical concept in mathematics.
Dissecting the Function: p(x) = √(x-1) + 2
Before we can pinpoint the domain, it's crucial to understand the function's structure. Our function, $p(x) = \sqrtx-1} + 2$, is a combination of two key mathematical operations$, is where we need to focus our attention when determining the domain. This is because the square root function has an inherent restriction: it cannot accept negative numbers as input. Taking the square root of a negative number results in an imaginary number, which falls outside the realm of real numbers that we typically deal with in basic function analysis. The '+ 2' part of the function is a simple vertical shift and doesn't impose any additional restrictions on the domain.
Therefore, our primary concern is ensuring that the expression inside the square root, which is (x-1), is greater than or equal to zero. This constraint forms the basis for determining the function's domain. We need to find all values of x that satisfy the inequality x - 1 ≥ 0. This inequality represents the condition under which the square root will produce a real number output. By solving this inequality, we will effectively define the domain of the function $p(x)$. The domain, in essence, will be the set of all x values that make the function “work” in the real number system. This careful consideration of the function's components and their limitations is essential for accurately determining the domain.
Identifying Restrictions: The Square Root's Constraint
As previously highlighted, the square root function is the primary source of restriction in our function, $p(x) = \sqrt{x-1} + 2$. The fundamental principle is that we cannot take the square root of a negative number within the realm of real numbers. This is a core mathematical rule that dictates the behavior of square root functions. Consequently, the expression inside the square root, x - 1, must be non-negative. This means it can be zero or any positive number. If x - 1 were to be negative, the result would be an imaginary number, which is not within the scope of the real-valued function we are analyzing.
To ensure a valid output from the square root, we establish the inequality x - 1 ≥ 0. This inequality mathematically expresses the condition that the expression inside the square root must be greater than or equal to zero. Solving this inequality will directly lead us to the domain of the function. It will tell us the range of x values that are permissible, those that will produce a real number output when plugged into the function. The '+ 2' in the function only shifts the graph vertically and does not affect the domain as it does not introduce any new restrictions on the possible x values. Therefore, our focus remains solely on the x - 1 term under the square root to determine the domain accurately.
Solving for the Domain: x - 1 ≥ 0
To determine the domain of the function $p(x) = \sqrt{x-1} + 2$, we need to solve the inequality x - 1 ≥ 0. This inequality represents the condition that the expression inside the square root must be non-negative. The process of solving this inequality is straightforward and involves isolating x on one side of the inequality. We begin by adding 1 to both sides of the inequality:
x - 1 + 1 ≥ 0 + 1
This simplifies to:
x ≥ 1
This result, x ≥ 1, is the key to understanding the function's domain. It tells us that the function is defined for all values of x that are greater than or equal to 1. In other words, we can input any value of x that is 1 or larger into the function, and we will get a real number output. If we were to input a value less than 1, the expression inside the square root would become negative, leading to an imaginary number, which is not allowed within the context of real-valued functions. Therefore, the solution to the inequality x ≥ 1 directly defines the domain of the function $p(x)$.
Expressing the Domain: Interval Notation
Now that we've solved the inequality and found that x ≥ 1, we need to express this solution in interval notation, which is a standard way of representing domains and ranges in mathematics. Interval notation uses brackets and parentheses to indicate the inclusion or exclusion of endpoints in an interval. A square bracket, [ or ], indicates that the endpoint is included in the interval, while a parenthesis, ( or ), indicates that the endpoint is excluded.
In our case, x ≥ 1 means that x can be 1 or any number greater than 1, extending infinitely in the positive direction. To represent this in interval notation, we use a square bracket to include 1 and the infinity symbol (∞) to represent the unbounded upper limit. Infinity is always enclosed in a parenthesis because it is not a specific number and cannot be included as an endpoint.
Therefore, the domain of the function $p(x) = \sqrt{x-1} + 2$ in interval notation is [1, ∞). This notation concisely conveys that the function is defined for all x values starting from 1 (inclusive) and extending to positive infinity. Understanding and using interval notation is crucial for accurately communicating domains and ranges in mathematical contexts.
The Correct Answer and Why
Based on our analysis, the domain of the function $p(x) = \sqrt{x-1} + 2$ is [1, ∞). This means that the correct answer is C. [1, ∞). Let's examine why the other options are incorrect:
- A. (-∞, 1]: This interval represents all numbers less than or equal to 1. However, our function requires x to be greater than or equal to 1, so this is incorrect.
- B. (-∞, 2]: This interval includes numbers less than 1, which would result in taking the square root of a negative number. Therefore, this is also incorrect.
- D. [2, ∞): While this interval includes numbers that are valid inputs for the function, it excludes the number 1, which is also a valid input. Therefore, this option is not the complete domain.
Option C, [1, ∞), is the only interval that accurately represents all possible input values for the function $p(x) = \sqrt{x-1} + 2$. It includes 1, as the expression inside the square root can be zero, and it extends to infinity, encompassing all values greater than 1. This thorough examination of the options reinforces the correctness of our identified domain.
Conclusion: Mastering Domain Determination
Determining the domain of a function is a fundamental skill in mathematics, and understanding the restrictions imposed by different mathematical operations is key to mastering this skill. In the case of the function $p(x) = \sqrt{x-1} + 2$, the square root function introduced the primary restriction, requiring the expression inside the square root to be non-negative. By setting up and solving the inequality x - 1 ≥ 0, we were able to accurately determine the domain as [1, ∞).
This exercise highlights the importance of careful analysis and attention to detail when working with functions. Recognizing potential restrictions and expressing the domain in correct notation are crucial for further mathematical explorations. The ability to confidently determine the domain of various functions lays the groundwork for more advanced concepts in calculus and analysis. By understanding the principles discussed in this article, you are well-equipped to tackle similar problems and deepen your understanding of functions and their properties.
