Domain ≥ 8: Which Function Fits?

by Alex Johnson 33 views

Determining the domain of a function is a fundamental concept in mathematics. The domain represents all possible input values (often x-values) for which a function is defined. In simpler terms, it's the set of numbers you can plug into a function and get a real number output. When dealing with square root functions, we need to be particularly mindful of the domain because we can't take the square root of a negative number and get a real number result. This restriction plays a crucial role in solving the problem at hand, where we're looking for a function with a specific domain: {xext∣extxext≥ext8x ext{ }| ext{ } x ext{ }≥ ext{ } 8}. This notation means we want a function that is only defined for x-values greater than or equal to 8. Let's break down why this constraint exists for square root functions and then examine the options provided to find the one that matches this domain. To truly understand this, it's essential to grasp the nature of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. However, if we try to take the square root of a negative number, like -9, we run into a problem. There's no real number that, when multiplied by itself, gives a negative result. This is because a positive number times a positive number is positive, and a negative number times a negative number is also positive. To deal with square roots of negative numbers, we venture into the realm of imaginary numbers, which involve the unit i, where i is defined as the square root of -1. But for functions that output real numbers, we must restrict the values under the square root to be non-negative (zero or positive). This is why the expression inside the square root, called the radicand, must be greater than or equal to zero. This restriction forms the basis for determining the domain of square root functions. When we have a function like f(x)=ext√(something)f(x) = ext{√}(something), the "something" has to be greater than or equal to zero for the function to produce a real output. This "something" could be a simple variable like x, or it could be a more complex expression involving x, constants, and other operations. The key is to set that entire expression greater than or equal to zero and then solve for x. The solution will give us the domain of the function. In the context of our problem, we are given several functions involving square roots, and we need to identify the one whose domain is {xext∣extxext≥ext8x ext{ }| ext{ } x ext{ }≥ ext{ } 8}. This means that the expression inside the square root must be greater than or equal to zero only when x is greater than or equal to 8. To find the correct function, we will analyze each option, focusing on the expression inside the square root and determining the values of x that make that expression non-negative.

Analyzing the Functions

Now, let's examine each function provided in the problem and determine its domain. This involves focusing on the radicand, the expression under the square root, and ensuring it is greater than or equal to zero. For each function, we will set up an inequality and solve for x. This will reveal the range of x-values for which the function is defined. This is a critical step in identifying the function that matches the given domain of {xext∣extxext≥ext8x ext{ }| ext{ } x ext{ }≥ ext{ } 8}. Remember, the domain is the set of all possible input values (x-values) that produce a real number output for the function. Square root functions are particularly sensitive to domain restrictions because we cannot take the square root of a negative number within the real number system. Therefore, the expression under the square root must always be zero or positive. Let's begin with the first function and systematically analyze each option.

A. f(x)=ext√(x−8)+1f(x) = ext{√}(x - 8) + 1

For the function f(x)=ext√(x−8)+1f(x) = ext{√}(x - 8) + 1, the expression under the square root is (x - 8). To find the domain, we set this expression greater than or equal to zero:

x−8≥0x - 8 ≥ 0

Adding 8 to both sides of the inequality, we get:

x≥8x ≥ 8

This means the function is defined for all x-values greater than or equal to 8. This perfectly matches the domain {xext∣extxext≥ext8x ext{ }| ext{ } x ext{ }≥ ext{ } 8} given in the problem. So, this function is a strong candidate, but we should still analyze the other options to be certain.

B. f(x)=ext√(x+8)−1f(x) = ext{√}(x + 8) - 1

Next, let's consider the function f(x)=ext√(x+8)−1f(x) = ext{√}(x + 8) - 1. The radicand in this case is (x + 8). Again, we set the expression under the square root greater than or equal to zero:

x+8≥0x + 8 ≥ 0

Subtracting 8 from both sides, we get:

x≥−8x ≥ -8

The domain of this function is {xext∣extxext≥ext−8x ext{ }| ext{ } x ext{ }≥ ext{ } -8}, which means the function is defined for all x-values greater than or equal to -8. This domain is different from the one specified in the problem (x≥8x ≥ 8), so this function is not the correct answer.

C. f(x)=ext√(x−1)+8f(x) = ext{√}(x - 1) + 8

Now, let's analyze the function f(x)=ext√(x−1)+8f(x) = ext{√}(x - 1) + 8. Here, the radicand is (x - 1). We set the expression under the square root greater than or equal to zero:

x−1≥0x - 1 ≥ 0

Adding 1 to both sides, we get:

x≥1x ≥ 1

The domain of this function is {xext∣extxext≥ext1x ext{ }| ext{ } x ext{ }≥ ext{ } 1}, which means the function is defined for all x-values greater than or equal to 1. This domain does not match the given domain of {xext∣extxext≥ext8x ext{ }| ext{ } x ext{ }≥ ext{ } 8}, so this function is not the correct answer.

D. f(x)=ext√(x+1)−8f(x) = ext{√}(x + 1) - 8

Finally, let's examine the function f(x)=ext√(x+1)−8f(x) = ext{√}(x + 1) - 8. The radicand for this function is (x + 1). We set the expression under the square root greater than or equal to zero:

x+1≥0x + 1 ≥ 0

Subtracting 1 from both sides, we get:

x≥−1x ≥ -1

The domain of this function is {xext∣extxext≥ext−1x ext{ }| ext{ } x ext{ }≥ ext{ } -1}, meaning the function is defined for all x-values greater than or equal to -1. This domain also does not match the target domain of {xext∣extxext≥ext8x ext{ }| ext{ } x ext{ }≥ ext{ } 8}, so this function is not the answer.

Conclusion

After carefully analyzing each function, we have determined that only one function has the domain {xext∣extxext≥ext8x ext{ }| ext{ } x ext{ }≥ ext{ } 8}. This was achieved by setting the expression under the square root (the radicand) greater than or equal to zero and solving for x. This process allowed us to identify the range of x-values for which each function is defined. Remember, the domain is a fundamental aspect of understanding functions in mathematics, and it's crucial to consider domain restrictions, especially when dealing with square roots, fractions, and logarithms. Based on our analysis, the correct answer is:

A. f(x)=ext√(x−8)+1f(x) = ext{√}(x - 8) + 1

This function is the only one that is defined for all x-values greater than or equal to 8, making it the perfect match for the given domain. For further learning on domains and functions, you can visit Khan Academy's article on domain and range.