Finding the Domain of a Function

To find the domain of the function \(f(x) = \frac{{\sqrt {x - 1} }}{{x - 4}}\), we need to identify all the real numbers \(x\) for which the function is defined. There are two main conditions we must consider for this specific function involving a square root and a denominator.

Conditions for the Domain

For the function \(f(x)\) to be defined in the set of real numbers, two conditions must be satisfied:

1. The expression under the square root must be non-negative.
2. The denominator cannot be equal to zero.

Condition 1: Expression Under Square Root

The term inside the square root is \((x - 1)\). For \(\sqrt{x - 1}\) to be a real number, we must have:

\(x - 1 \ge 0\)

Adding 1 to both sides of the inequality, we get:

\(x \ge 1\)

This means that \(x\) must be greater than or equal to 1.

Condition 2: Denominator Cannot Be Zero

The denominator of the function is \((x - 4)\). The function is undefined when the denominator is zero. Therefore, we must have:

\(x - 4 \ne 0\)

Adding 4 to both sides of the inequality, we get:

\(x \ne 4\)

This means that \(x\) cannot be equal to 4.

Combining the Conditions

To find the domain of \(f(x)\), we must satisfy both conditions simultaneously: \(x \ge 1\) and \(x \ne 4\).

The condition \(x \ge 1\) corresponds to the interval \([1, \infty)\) in interval notation. This includes all numbers starting from 1 and extending infinitely to the right, including 1 itself.

From this set \([1, \infty)\), we must exclude the value \(x = 4\). The number 4 is indeed included in the interval \([1, \infty)\) because \(4 \ge 1\).

To exclude 4 from the interval \([1, \infty)\), we split the interval at 4. The values \(x\) that satisfy \(x \ge 1\) and \(x \ne 4\) are those where:

• \(x\) is greater than or equal to 1 AND less than 4, OR
• \(x\) is greater than 4.

In interval notation, this is represented as the union of two intervals:

• \([1, 4)\) represents the numbers \(x\) such that \(1 \le x < 4\). The square bracket \([\) includes 1, and the parenthesis \()\) excludes 4.
• \((4, \infty)\) represents the numbers \(x\) such that \(x > 4\). The parenthesis \(()\) excludes 4, and \(\infty\) is always excluded.

Combining these two intervals gives the domain of the function \(f(x)\):

\([1, 4) \cup (4, \infty)\)

Comparing with Options

Let's look at the given options:

1. \((-\infty, 4) \cup (4, \infty)\): This includes all real numbers except 4, but it does not satisfy the condition \(x \ge 1\).
2. \((4, \infty)\): This satisfies \(x \ne 4\) and \(x > 4\), which implies \(x \ge 1\), but it excludes values like \(x=1, 2, 3\), which are in the domain.
3. \((1, 4) \cup (4, \infty)\): This satisfies \(x \ne 4\) and includes values \(x > 1\). However, it excludes \(x=1\), which is in the domain because \(\sqrt{1-1}/(1-4) = \sqrt{0}/(-3) = 0/-3 = 0\), which is a defined real number.
4. \([1, 4) \cup (4, \infty)\): This includes values \(x\) such that \(1 \le x < 4\) or \(x > 4\). This satisfies both \(x \ge 1\) and \(x \ne 4\).

The domain we found, \([1, 4) \cup (4, \infty)\), matches option 4.

Revision Table: Domain of Functions

Additional Information: Interval Notation

Interval notation is a way to express sets of real numbers using parentheses and square brackets.

• Parentheses \((a, b)\) indicate an open interval, meaning the endpoints \(a\) and \(b\) are not included.
• Square brackets \([a, b]\) indicate a closed interval, meaning the endpoints \(a\) and \(b\) are included.
• A combination like \([a, b)\) or \((a, b]\) is a half-open or half-closed interval, where one endpoint is included and the other is not.
• The symbols \(\infty\) (infinity) and \(-\infty\) (negative infinity) are always used with parentheses because infinity is a concept, not a specific number that can be included.
• The union symbol \(\cup\) is used to combine two or more intervals. For example, \((-\infty, a) \cup (a, \infty)\) represents all real numbers except \(a\).