Verified Answer :

  • The domain of a function \(\sqrt{g(x)}\) requires the radicand to be non-negative: \(g(x) \geq 0\).
  • For \(f(x) = \sqrt{x - 30}\), set up the inequality \(x - 30 \geq 0\).
  • Solve for \(x\) to find the domain: \(x \geq 30\).
  • The final answer is \(\boxed{30}\).

Explanation

  1. Condition for a Defined Square Root Function

The domain of a square root function of the form \(f(x) = \sqrt{g(x)}\) consists of all real numbers \(x\) such that the expression under the square root, called the radicand, is non-negative. This is because the square root of a negative number is not a real number. Therefore, to determine the domain, we must ensure that the radicand is greater than or equal to zero.

  1. Setting up the Inequality

For the function \(f(x) = \sqrt{x - 30}\), the radicand is \(x - 30\). According to our condition, we must have:
\[x - 30 \geq 0\]

  1. Solving the Inequality and Final Conclusion

To solve the inequality \(x - 30 \geq 0\), we add \(30\) to both sides of the inequality:
\[x \geq 30\]
This result tells us that the function is defined for all values of \(x\) greater than or equal to \(30\). Thus, the domain is the interval \([30, \infty)\).

Examples

In physical applications, square root functions often represent thresholds or physical constraints. For instance, if a machine only operates safely when the temperature \(x\) (in degrees Celsius) is at least 30, the function \(f(x) = \sqrt{x - 30}\) might represent a safety sensor output that produces a real-valued signal only when the system is running at or above its minimum operational temperature. Identifying the domain helps technicians understand the operational limits of the equipment.

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