camdengibson89
Answered

Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

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

The domain of the function is [tex]\( \square \leq x \leq \square \)[/tex].

Sagot :

GinnyAnswer
To determine the domain of the function [tex]\( f(x) = 2 \sqrt{-x^2 + 10x} \)[/tex], we need to ensure that the expression inside the square root is non-negative. This is because the square root of a negative number is not defined in the real number system.

1. Start with the expression inside the square root: [tex]\(-x^2 + 10x \geq 0\)[/tex].

2. Factor the quadratic expression: [tex]\(-x^2 + 10x = 10x - x^2 = x(10 - x)\)[/tex].

3. Set up the inequality: [tex]\(x(10 - x) \geq 0\)[/tex].

4. Determine the values of [tex]\(x\)[/tex] that satisfy this inequality. The expression [tex]\(x(10 - x)\)[/tex] will be zero at [tex]\(x = 0\)[/tex] and [tex]\(x = 10\)[/tex].

5. Analyze the sign of the expression [tex]\(x(10 - x)\)[/tex] within the interval [tex]\([0, 10]\)[/tex]:
- For [tex]\(x\)[/tex] in [tex]\(0 \leq x \leq 10\)[/tex], the expression [tex]\(x(10 - x)\)[/tex] is non-negative because it represents the product of two non-negative numbers within this interval.
- Outside this interval, the expression [tex]\(x(10 - x)\)[/tex] becomes negative because one or both factors would be negative.

Thus, the function [tex]\( f(x) \)[/tex] is defined for [tex]\( x \)[/tex] in the interval [tex]\([0, 10]\)[/tex].

Hence, the domain of the function is [tex]\( 0 \leq x \leq 10 \)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.