91Ó°ÊÓ

The square root function is a fundamental concept that appears frequently in various mathematical contexts. It is represented as \( \sqrt{x} \), where \( x \) is the number or expression whose square root is to be taken. In this case, our focus is on the function \( k(x) = \frac{\sqrt{x+3}}{x-1} \).
The expression under the square root, \( x+3 \), must be non-negative to ensure that the square root is real and defined. This is because the square root of a negative number is not a real number. Hence, we set \( x+3 \geq 0 \).
After simplifying, this gives us \( x \geq -3 \). This inequality is one of the key considerations when finding the domain of the function, ensuring that all values of \( x \) meet this condition to make the square root function valid.

In any mathematical expression involving division, as in our function \( k(x) = \frac{\sqrt{x+3}}{x-1} \), it's critical to remember that division by zero is undefined. This means that the denominator, which is \( x-1 \) in this case, should never be equal to zero. If it were, it would make the function undefined at that point.
To prevent undefined situations, we solve the equation \( x-1 = 0 \). Solving this, we find \( x = 1 \). Thus, \( x eq 1 \) for our function's domain.
This restriction is crucial because any value of \( x \) that makes the denominator zero must be excluded from the domain to maintain a valid and defined function.

Interval notation is a concise way to describe the set of all numbers that meet certain conditions. After finding the conditions for the function \( k(x) = \frac{\sqrt{x+3}}{x-1} \), we can express its domain in interval notation.
From the previous sections, we determined:

• \( x \geq -3 \) to ensure a defined square root function.
• \( x eq 1 \) to avoid undefined division.

When combining these conditions, we include all values \( x \geq -3 \), except \( x = 1 \). Using interval notation, the domain is written as \([-3, 1) \cup (1, \infty)\). This representation uses brackets and parentheses to clearly indicate the intervals of valid \( x \) values, enhancing understanding of how the function behaves across its domain.