What is the domain of the function \(h(x) = \frac{1}{x - 6}\)?

What is the domain of the function \(h(x) = \frac{1}{x - 6}\)?

• A. All real numbers except x = 6
• B. All real numbers
• C. x ≥ 6
• D. x ≤ 0

Answer: (A)

The domain of the function \(h(x) = \frac{1}{x - 6}\) consists of all the values of \(x\) for which the function is defined. Since the function is a rational expression, it is crucial to identify any values that would make the denominator zero. The denominator in this case is \(x - 6\). When \(x - 6 = 0\), solving this gives \(x = 6\). At this point, the function becomes undefined because division by zero is not permitted in mathematics. Therefore, the value \(x = 6\) must be excluded from the domain. Thus, the domain of the function is all real numbers except \(x = 6\). This captures every real number that can be substituted into the function without causing the denominator to equal zero. Since the only restriction is that \(x\) cannot equal 6, the correct representation of the domain is indeed all real numbers except for this particular value.

The domain of the function (h(x) = \frac{1}{x - 6}) consists of all the values of (x) for which the function is defined. Since the function is a rational expression, it is crucial to identify any values that would make the denominator zero. The denominator in this case is (x - 6).

When (x - 6 = 0), solving this gives (x = 6). At this point, the function becomes undefined because division by zero is not permitted in mathematics. Therefore, the value (x = 6) must be excluded from the domain.

Thus, the domain of the function is all real numbers except (x = 6). This captures every real number that can be substituted into the function without causing the denominator to equal zero. Since the only restriction is that (x) cannot equal 6, the correct representation of the domain is indeed all real numbers except for this particular value.