Carmen's Age: A Mathematical Puzzle

Hey guys! Today, we're diving into a fun math problem that will help us determine Carmen's age. It's like a little detective work with numbers! We need to figure out the domain of a combined function and then use that information to unlock the secret of Carmen's age. Ready to put on your thinking caps?

The Challenge: Unveiling Carmen's Age

The core of the puzzle lies in understanding the functions f(x) = √x and g(x) = √1-x. Our mission is to find the domain of their product, which we'll call (∙)(x). This means identifying all the possible values of 'x' that make the function work without causing any mathematical mayhem (like taking the square root of a negative number!). Once we've nailed down the domain, we'll use it to calculate a special sum, multiply it by three, and voilà, we'll have Carmen's age!

Delving into the Domain of Functions

Let's break down what the domain really means. Think of a function like a machine: you feed it an input ('x'), and it spits out an output (f(x) or g(x) in this case). The domain is simply the set of all possible inputs that the machine can handle without breaking down. For square root functions, like the ones we're dealing with, the golden rule is that we can't take the square root of a negative number (at least not in the realm of real numbers!). This is because there's no real number that, when multiplied by itself, gives you a negative result. So, our inputs must ensure that the expressions under the square roots are either zero or positive.

For f(x) = √x, this means that x must be greater than or equal to 0. We can write this mathematically as x ≥ 0. Any negative value for x would lead to the square root of a negative number, which is a no-go. For example, if x = -1, then f(x) = √-1, which is not a real number.

Now, let's consider g(x) = √1-x. Here, the expression under the square root is 1-x. To avoid taking the square root of a negative number, we need 1-x to be greater than or equal to 0. This gives us the inequality 1-x ≥ 0. If we rearrange this inequality, we get 1 ≥ x, or equivalently, x ≤ 1. This means that x must be less than or equal to 1. If x is greater than 1, then 1-x would be negative, and we'd be back to the square root of a negative number. For example, if x = 2, then g(x) = √1-2 = √-1, which is not a real number.

Unveiling the Domain of (∙)(x)

Now, the fun part! We need to find the domain of the function (∙)(x), which is the product of f(x) and g(x). In other words, (∙)(x) = f(x) * g(x) = √x * √1-x. For this combined function to work, both f(x) and g(x) must be defined. This means that x must satisfy both x ≥ 0 (from f(x)) and x ≤ 1 (from g(x)).

Think of it like a Venn diagram. We have two conditions: x must be in the set of numbers greater than or equal to 0, and x must be in the set of numbers less than or equal to 1. The domain of (∙)(x) is the intersection of these two sets, the values of x that satisfy both conditions simultaneously. The numbers that fit this bill are all the numbers between 0 and 1, inclusive. We can write this as 0 ≤ x ≤ 1. This is the domain of (∙)(x).

Cracking the Code: The Sum of Integer Values

Our next step is to find the sum of the integer values within the domain we just found. Remember, integers are whole numbers (no fractions or decimals). In the domain 0 ≤ x ≤ 1, the integers are simply 0 and 1. So, the sum of the integer values in the domain of (∙)(x) is 0 + 1 = 1. This is a crucial piece of information in our quest to uncover Carmen's age.

The Final Calculation: Carmen's Age Revealed

The problem states that Carmen's age is three times the sum of the integer values we just calculated. We found the sum to be 1, so Carmen's age is 3 * 1 = 3 years. And there you have it! We've successfully navigated the mathematical landscape, conquered the domain challenge, and revealed Carmen's age.

The Answer: Carmen is 3 Years Old

So, the correct answer is C) 3 años. We did it, guys! By carefully analyzing the functions, understanding the concept of domains, and performing a few simple calculations, we've solved the puzzle and discovered Carmen's age. Math can be pretty cool, right?

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Domain of a Function: The Foundation of Our Puzzle

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