- The function \(f(x) = 2 \sqrt{-x^2 + 10x}\) requires \(-x^2 + 10x \geq 0\) for real outputs.
- Factoring gives \(x(10 - x) \geq 0\), which is true for \(x \in [0, 10]\).
- The domain of the function is \(\boxed{0} \leq x \leq \boxed{10}\).
Explanation
- Defining the Domain Condition
The function \(f(x) = 2 \sqrt{-x^2 + 10x}\) is defined for real values of \(x\) only if the expression inside the square root is non-negative. This is because the square root of a negative number is not a real number. Therefore, we must satisfy the condition:
\[-x^2 + 10x \geq 0\]
- Solving the Inequality
To find the values of \(x\) that satisfy the inequality \(-x^2 + 10x \geq 0\), we first factor the quadratic expression:
\[-x(x - 10) \geq 0\]
Multiplying both sides by \(-1\) reverses the inequality sign:
\[x(x - 10) \leq 0\]
This quadratic inequality is satisfied when \(x\) lies between its roots. The roots of the equation \(x(x - 10) = 0\) are \(x = 0\) and \(x = 10\). Testing a value in the interval \([0, 10]\), such as \(x = 5\), we find \(5(5 - 10) = -25 \leq 0\), which confirms that the interval \([0, 10]\) is the solution set.
- Conclusion
Since the inequality \(-x^2 + 10x \geq 0\) holds for all \(x\) in the interval \([0, 10]\), the domain of the function \(f(x)\) is the set of all \(x\) such that \(0 \leq x \leq 10\). Therefore, the lower bound is \(0\) and the upper bound is \(10\).
The domain is \(\boxed{0} \leq x \leq \boxed{10}\).
Examples
In architecture and engineering, determining the domain of a function is crucial when designing structures like domes. If a dome's height \(f(x)\) is given by a function \(f(x) = 2 \sqrt{-x^2 + 10x}\), the domain tells the engineer the exact range of horizontal distance \(x\) where the dome exists. Knowing that the structure spans from \(x=0\) to \(x=10\) allows them to calculate the base diameter, total material required, and ensure the structural integrity of the build. Mathematically defining this range ensures that physical dimensions stay within real, measurable limits.