Find The Domain Of The Function \[$ K \$\] Where \[$ K(x) = -\frac{7}{\sqrt{x-6}} \$\].If You Need To Enter \[$\infty\$\], You Can Type Inf Or Infinity. If You Need To Union Two (or More) Intervals, Use The Letter \[$ U
Introduction
In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function K(x) is valid. In this article, we will explore the domain of the function K(x) = -\frac{7}{\sqrt{x-6}}.
Understanding the Function
The function K(x) = -\frac{7}{\sqrt{x-6}} is a rational function, which means it is the ratio of two functions. The numerator of the function is a constant, -7, while the denominator is the square root of (x-6). The square root function is defined only for non-negative values, so we must ensure that the expression inside the square root is non-negative.
Finding the Domain
To find the domain of the function K(x), we need to determine the values of x for which the expression inside the square root is non-negative. In other words, we need to find the values of x for which x-6 ≥ 0.
Solving the Inequality
To solve the inequality x-6 ≥ 0, we can add 6 to both sides of the inequality, which gives us x ≥ 6.
Understanding the Result
The result of the inequality x ≥ 6 tells us that the expression inside the square root is non-negative for all values of x greater than or equal to 6. However, we must also consider the fact that the denominator of the function cannot be zero. Since the denominator is the square root of (x-6), we must ensure that x-6 > 0, which gives us x > 6.
Combining the Results
Combining the results of the inequality x ≥ 6 and the fact that x > 6, we can conclude that the domain of the function K(x) is all real numbers greater than 6.
Conclusion
In conclusion, the domain of the function K(x) = -\frac{7}{\sqrt{x-6}} is all real numbers greater than 6. This means that the function is defined for all values of x greater than 6, and it is not defined for any values of x less than or equal to 6.
Domain of K(x)
The domain of K(x) is (6, ∞).
Final Answer
Introduction
In our previous article, we explored the domain of the function K(x) = -\frac{7}{\sqrt{x-6}}. In this article, we will answer some frequently asked questions about the domain of a function.
Q: What is the domain of a function?
A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function is valid.
Q: How do I find the domain of a function?
A: To find the domain of a function, you need to determine the values of x for which the function is defined. This may involve solving inequalities, checking for restrictions on the domain, and considering any discontinuities in the function.
Q: What are some common restrictions on the domain of a function?
A: Some common restrictions on the domain of a function include:
- Division by zero: The function is not defined when the denominator is zero.
- Square root of a negative number: The function is not defined when the expression inside the square root is negative.
- Logarithm of a non-positive number: The function is not defined when the argument of the logarithm is non-positive.
Q: How do I represent the domain of a function?
A: The domain of a function can be represented in several ways, including:
- Interval notation: The domain is represented as an interval, such as (a, b) or [a, b].
- Union of intervals: The domain is represented as a union of intervals, such as (a, b) U (c, d).
- Set notation: The domain is represented as a set of values, such as {x | x > 0}.
Q: What is the difference between the domain and the range of a function?
A: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values. In other words, the domain is the set of all possible x-values, while the range is the set of all possible y-values.
Q: Can the domain of a function be empty?
A: Yes, the domain of a function can be empty. This occurs when the function is not defined for any values of x.
Q: Can the domain of a function be infinite?
A: Yes, the domain of a function can be infinite. This occurs when the function is defined for all real numbers, or for all positive integers, or for all negative integers.
Q: How do I find the domain of a composite function?
A: To find the domain of a composite function, you need to find the domain of each individual function and then intersect the domains. In other words, you need to find the values of x for which both functions are defined.
Q: What is the domain of the function f(x) = \frac{1}{x-2}?
A: The domain of the function f(x) = \frac{1}{x-2} is all real numbers except x = 2. This is because the function is not defined when the denominator is zero.
Q: What is the domain of the function g(x) = \sqrt{x+3}?
A: The domain of the function g(x) = \sqrt{x+3} is all real numbers greater than or equal to -3. This is because the function is not defined when the expression inside the square root is negative.
Conclusion
In conclusion, the domain of a function is the set of all possible input values for which the function is defined. Finding the domain of a function involves determining the values of x for which the function is defined, and considering any restrictions on the domain. By understanding the domain of a function, you can better understand the behavior of the function and make more informed decisions about its use.