Finding The Domain Of G(x) = √(x-2) / (x-5)

In mathematics, determining the domain of a function is a crucial step in understanding its behavior and properties. The domain of a function is the set of all possible input values (often denoted as x) for which the function produces a valid output. When dealing with functions that involve square roots and fractions, there are specific restrictions on the input values that we need to consider.

Understanding Domain Restrictions

To effectively find the domain of a function, it's essential to understand the restrictions imposed by different mathematical operations. Here are the two primary restrictions we'll focus on in this article:

1. Square Roots: The expression inside a square root (the radicand) must be greater than or equal to zero. This is because the square root of a negative number is not a real number.
2. Fractions: The denominator of a fraction cannot be equal to zero. Division by zero is undefined in mathematics.

In the given problem, we have the function:

$g(x) = \frac{\sqrt{x-2}}{x-5}$

This function combines both a square root and a fraction, so we must consider both restrictions to determine its domain. Let's delve into each restriction step by step.

Square Root Restriction: Ensuring Non-Negative Radicand

For the square root portion of the function, $\sqrt{x-2}$, we need to ensure that the radicand, x - 2, is greater than or equal to zero. Mathematically, this can be expressed as:

$x - 2 \geq 0$

To solve this inequality, we add 2 to both sides:

$x \geq 2$

This inequality tells us that x must be greater than or equal to 2 for the square root to produce a real number output. In interval notation, this condition can be represented as $[2, \infty)$. This means the domain includes all real numbers from 2 (inclusive) to positive infinity. However, we must also consider the restriction imposed by the fraction in the function.

Fraction Restriction: Avoiding Division by Zero

Now, let's address the fractional part of the function, $\frac{\sqrt{x-2}}{x-5}$. The denominator of this fraction is x - 5. To avoid division by zero, we must ensure that:

$x - 5 \neq 0$

Adding 5 to both sides, we get:

$x \neq 5$

This inequality indicates that x cannot be equal to 5. If x were 5, the denominator would become zero, and the function would be undefined.

Combining Restrictions: Determining the Overall Domain

We've established two crucial restrictions on the domain of g(x):

1. x ≥ 2 (from the square root)
2. x ≠ 5 (from the fraction)

To find the overall domain, we need to consider both of these restrictions simultaneously. The first restriction tells us that x must be in the interval $[2, \infty)$. The second restriction tells us that x cannot be 5. Therefore, we need to exclude 5 from the interval $[2, \infty)$.

We can represent this exclusion using interval notation. The domain will consist of two intervals: the interval from 2 to 5 (excluding 5) and the interval from 5 to infinity. In interval notation, this is written as:

$[2, 5) \cup (5, \infty)$

This notation means that the domain of g(x) includes all real numbers greater than or equal to 2, except for 5. The parenthesis around 5 indicate that 5 is not included in the domain.

Visualizing the Domain on a Number Line

A helpful way to visualize the domain is by using a number line. Draw a number line and mark the critical points: 2 and 5.

• Since x must be greater than or equal to 2, we draw a closed bracket at 2, indicating that 2 is included in the domain. Shade the region to the right of 2, representing all values greater than 2.
• Since x cannot be equal to 5, we draw an open parenthesis at 5, indicating that 5 is not included in the domain. There will be a "hole" at 5, visually representing the exclusion.
• The shaded region to the right of 2, excluding 5, represents the domain of the function.

This visual representation reinforces the interval notation $[2, 5) \cup (5, \infty)$.

Writing the Domain in Set Notation

While interval notation is a common and concise way to represent the domain, we can also express it using set notation. Set notation provides a more formal way to describe the set of all possible input values.

The domain of g(x) in set notation can be written as:

$\lbrace x \in \mathbb{R} \mid x \geq 2 \text{ and } x \neq 5 \rbrace$

This notation reads as "the set of all x belonging to the set of real numbers such that x is greater than or equal to 2 and x is not equal to 5." This is a precise and unambiguous way to define the domain of the function.

Example Problems: Applying Domain Restrictions

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

Example 1: Find the domain of $f(x) = \frac{1}{\sqrt{x+3}}$

• Square Root Restriction: $x + 3 > 0$ (Note that we use > instead of ≥ because the square root is in the denominator)
• Solving for x: $x > -3$
• Fraction Restriction: The denominator, $\sqrt{x+3}$, cannot be zero. This is already addressed by the square root restriction, which requires $x + 3$ to be strictly greater than zero.
• Domain: $(-3, \infty)$

Example 2: Find the domain of $h(x) = \sqrt{4-x^2}$

• Square Root Restriction: $4 - x^2 \geq 0$
• Rearranging: $x^2 \leq 4$
• Taking the square root of both sides: $|x| \leq 2$
• This inequality means $-2 \leq x \leq 2$
• Domain: $[-2, 2]$

These examples illustrate the importance of carefully considering all restrictions imposed by the function's components when determining its domain.

Key Takeaways for Finding Domains

• Identify Restrictions: When finding the domain of a function, first identify any restrictions imposed by operations like square roots, fractions, logarithms, etc.
• Square Roots: The expression inside a square root must be greater than or equal to zero.
• Fractions: The denominator of a fraction cannot be equal to zero.
• Solve Inequalities: Set up and solve the appropriate inequalities to determine the valid input values.
• Combine Restrictions: If a function has multiple restrictions, combine them to find the overall domain.
• Use Interval Notation: Express the domain using interval notation for a clear and concise representation.
• Visualize on a Number Line: A number line can be a helpful tool for visualizing the domain and understanding the restrictions.

Conclusion: Mastering Domain Determination

Finding the domain of a function is a fundamental skill in mathematics. By understanding the restrictions imposed by various mathematical operations and applying the techniques discussed in this article, you can confidently determine the domain of a wide range of functions. Remember to always consider square roots, fractions, and any other potential restrictions to ensure you've identified all the valid input values. Mastering this skill will provide a solid foundation for further exploration of functions and their properties.

Therefore, the domain of the function $g(x) = \frac{\sqrt{x-2}}{x-5}$ is $[2, 5) \cup (5, \infty)$.