Domain Of The Function P(x) = √(x-1) + 2 A Comprehensive Guide

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In mathematics, understanding the domain of a function is crucial for analyzing its behavior and properties. The domain of a function 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 causing any mathematical errors or undefined results. This article delves into the process of determining the domain of the function $p(x) = \sqrt{x-1} + 2$, providing a step-by-step explanation and ensuring a clear understanding of the underlying concepts.

Defining the Domain of a Function

Before we dive into the specifics of the function $p(x)$, let's establish a firm understanding of what the domain of a function truly means. The domain is the set of all real numbers x for which the function yields a real number output. Certain mathematical operations impose restrictions on the input values. The most common restrictions arise from:

  1. Division by zero: A function cannot be evaluated at a value that makes the denominator zero.
  2. Square roots of negative numbers: In the realm of real numbers, we cannot take the square root of a negative number.
  3. Logarithms of non-positive numbers: Logarithms are only defined for positive arguments.

When determining the domain of a function, we must identify any such restrictions and exclude the corresponding x-values from the domain. The domain is often expressed in interval notation, which provides a concise way to represent a range of values.

Analyzing the Function $p(x) = \sqrt{x-1} + 2$

Now, let's focus on our function: $p(x) = \sqrt{x-1} + 2$. This function involves a square root, which is a critical point to consider when determining the domain. As we mentioned earlier, the square root of a negative number is not a real number. Therefore, the expression inside the square root, x - 1, must be greater than or equal to zero.

This leads us to the inequality:

x−1≥0x - 1 \geq 0

To solve for x, we add 1 to both sides of the inequality:

x≥1x \geq 1

This inequality tells us that the function $p(x)$ is only defined for x-values that are greater than or equal to 1. In other words, we can only plug in values of x that are 1 or larger into the function and obtain a real number output. Any value of x less than 1 would result in taking the square root of a negative number, which is undefined in the real number system.

Therefore, the domain of the function $p(x) = \sqrt{x-1} + 2$ consists of all real numbers x such that x is greater than or equal to 1.

Expressing the Domain in Interval Notation

To express the domain in interval notation, we use brackets and parentheses to indicate the inclusion or exclusion of endpoints. A square bracket [ ] indicates that the endpoint is included in the interval, while a parenthesis ( ) indicates that the endpoint is excluded. Infinity (∞\infty) and negative infinity (−∞-\infty) are always enclosed in parentheses because they are not actual numbers but rather represent unboundedness.

In our case, the domain includes 1 and all numbers greater than 1, extending infinitely to the right. Therefore, the domain of $p(x)$ in interval notation is:

[1,∞)[1, \infty)

The square bracket on the left side indicates that 1 is included in the domain, and the parenthesis on the right side indicates that infinity is not a specific endpoint but rather a concept of unboundedness.

Visualizing the Domain on a Number Line

A number line provides a visual representation of the domain. To represent the domain of $p(x)$ on a number line, we would draw a closed circle (or a filled-in dot) at 1 to indicate that it is included in the domain. Then, we would draw a line extending to the right, with an arrowhead indicating that the domain continues indefinitely in the positive direction.

Why is Understanding the Domain Important?

Determining the domain of a function is not just a mathematical exercise; it has practical implications in various fields. Understanding the domain helps us:

  1. Identify valid inputs: Knowing the domain allows us to determine which input values will produce meaningful outputs. For example, in a function modeling the height of an object over time, negative time values might not be part of the domain.
  2. Avoid errors: By restricting the input values to the domain, we can prevent mathematical errors such as division by zero or taking the square root of a negative number.
  3. Interpret results correctly: The domain provides context for interpreting the output of a function. For instance, if the domain of a function is restricted to positive values, we know that the output only applies to positive inputs.
  4. Graph functions accurately: The domain is essential for graphing functions accurately. We only plot the function for x-values within its domain.

Common Mistakes to Avoid

When determining the domain of a function, it's crucial to avoid common mistakes. Here are a few to watch out for:

  1. Forgetting restrictions: Always remember to consider all potential restrictions, such as division by zero, square roots of negative numbers, and logarithms of non-positive numbers.
  2. Incorrectly solving inequalities: When an inequality is involved, make sure to solve it correctly to determine the proper range of x-values.
  3. Misinterpreting interval notation: Pay attention to the use of brackets and parentheses to accurately represent the inclusion or exclusion of endpoints.
  4. Ignoring context: In real-world applications, consider the context of the problem to determine if there are any additional restrictions on the domain.

Practice Problems

To solidify your understanding of domains, let's work through a few practice problems.

Problem 1: Find the domain of the function $f(x) = \frac{1}{x-2}$.

Solution: This function involves division. The denominator, x - 2, cannot be equal to zero. Therefore, x cannot be equal to 2. The domain is all real numbers except 2, which can be expressed in interval notation as $(-\infty, 2) \cup (2, \infty)$.

Problem 2: Find the domain of the function $g(x) = \sqrt{4-x}$.

Solution: This function involves a square root. The expression inside the square root, 4 - x, must be greater than or equal to zero. Solving the inequality 4 - x ≥ 0, we get x ≤ 4. The domain is all real numbers less than or equal to 4, which can be expressed in interval notation as $(-\infty, 4]$.

Problem 3: Find the domain of the function $h(x) = \ln(x+3)$.

Solution: This function involves a natural logarithm. The argument of the logarithm, x + 3, must be greater than zero. Solving the inequality x + 3 > 0, we get x > -3. The domain is all real numbers greater than -3, which can be expressed in interval notation as $(-3, \infty)$.

Conclusion

Determining the domain of a function is a fundamental concept in mathematics. By understanding the restrictions imposed by various mathematical operations and practicing problem-solving, you can confidently identify the domain of any function. In the case of $p(x) = \sqrt{x-1} + 2$, the domain is $[1, \infty)$, representing all real numbers greater than or equal to 1. Mastering this concept is crucial for further exploration of functions and their applications in diverse fields.