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The function, [tex]f(x)[/tex], describes the height of a dome on top of a building, where [tex]f(x)[/tex] is the height from the base of the dome and [tex]x[/tex] is the horizontal distance from where the dome meets the building.

[tex]f(x)=2 \sqrt{-x^2+10 x}[/tex]

The domain of the function is $\square \leq x \leq$ $\square$ .

The function is f ( x ) = 2 − x 2 + 10 x ​ .
To find the domain, set the expression inside the square root to be non-negative: − x 2 + 10 x ≥ 0 .
Factor the expression: x ( − x + 10 ) ≥ 0 .
Solve for x : 0 ≤ x ≤ 10 . The domain is 0 ≤ x ≤ 10 ​ .

Explanation

Understanding the Problem We are given the function f ( x ) = 2 − x 2 + 10 x ​ which describes the height of a dome. We need to find the domain of this function, which means we need to find the values of x for which the function is defined. Since we have a square root, the expression inside the square root must be non-negative.

Setting up the Inequality To find the domain, we need to solve the inequality − x 2 + 10 x ≥ 0 . We can factor out an x from the expression: x ( − x + 10 ) ≥ 0 .

Finding the Roots Now, we need to find the intervals where this inequality holds. The roots of the quadratic expression are x = 0 and − x + 10 = 0 ⇒ x = 10 . So the roots are x = 0 and x = 10 .

Testing the Intervals We will test the intervals ( − ∞ , 0 ) , ( 0 , 10 ) , and ( 10 , ∞ ) to see where the inequality x ( − x + 10 ) ≥ 0 is satisfied.

For x < 0 , let's take x = − 1 . Then ( − 1 ) ( − ( − 1 ) + 10 ) = ( − 1 ) ( 11 ) = − 11 , which is not greater than or equal to 0.

For 0 < x < 10 , let's take x = 5 . Then ( 5 ) ( − 5 + 10 ) = ( 5 ) ( 5 ) = 25 , which is greater than or equal to 0.

For 10"> x > 10 , let's take x = 11 . Then ( 11 ) ( − 11 + 10 ) = ( 11 ) ( − 1 ) = − 11 , which is not greater than or equal to 0.


Thus, the inequality is satisfied when 0 ≤ x ≤ 10 .

Determining the Domain Therefore, the domain of the function is 0 ≤ x ≤ 10 .

Final Answer The domain of the function f ( x ) = 2 − x 2 + 10 x ​ is 0 ≤ x ≤ 10 .


Examples
Understanding the domain of a function is crucial in various real-world applications. For instance, when designing a bridge with a parabolic arch, engineers need to determine the valid horizontal distances (x-values) for which the arch's height (f(x)) is physically possible. Similarly, in projectile motion, knowing the domain helps calculate the range of distances a projectile can travel, ensuring safety and accuracy in fields like sports or military applications. By defining the limits within which a function operates, we ensure that our models and designs are both realistic and functional.

Answered by GinnyAnswer | 2025-07-03

The domain of the function f ( x ) = 2 − x 2 + 10 x ​ is the interval 0 ≤ x ≤ 10 , meaning that the function is defined for values of x between 0 and 10, inclusive. To find this, we set the expression inside the square root to be non-negative and determined the critical points. Testing intervals revealed where the inequality holds true, leading to this domain.
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Answered by Anonymous | 2025-07-04