What Is The Domain Of The Function $f(x) = \sqrt{x+6}$ In Interval Notation?A. $[-6, \infty)$ B. $ ( 0 , ∞ ) (0, \infty) ( 0 , ∞ ) [/tex] C. $(-6, \infty)$ D. $[0, \infty)$
Introduction
When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. In this article, we'll explore the domain of the function $f(x) = \sqrt{x+6}$ and determine the correct interval notation.
What is the Domain of a Function?
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. For example, consider the function $f(x) = \frac{1}{x}$. The domain of this function is all real numbers except for 0, because dividing by zero is undefined.
The Square Root Function
The square root function, denoted by $\sqrt{x}$, is a fundamental function in mathematics. It's defined as the inverse of the square function, which raises a number to the power of 2. The square root function is only defined for non-negative real numbers, because the square of any real number is always non-negative.
The Function $f(x) = \sqrt{x+6}$
Now, let's consider the function $f(x) = \sqrt{x+6}$. This function is a variation of the square root function, with an added constant of 6. To determine the domain of this function, we need to consider the values of x for which the expression inside the square root is non-negative.
Determining the Domain
To determine the domain of the function $f(x) = \sqrt{x+6}$, we need to find 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 $x+6 \geq 0$.
Solving the Inequality
To solve the inequality $x+6 \geq 0$, we can subtract 6 from both sides, resulting in $x \geq -6$. This means that the expression inside the square root is non-negative for all values of x greater than or equal to -6.
Conclusion
Based on our analysis, we can conclude that the domain of the function $f(x) = \sqrt{x+6}$ is all real numbers greater than or equal to -6. In interval notation, this can be written as $[-6, \infty)$.
Answer
The correct answer is:
- A. $[-6, \infty)$
Final Thoughts
In conclusion, the domain of a function is the set of all possible input values (x-values) for which the function is defined. By understanding the concept of the domain, we can determine the values of x for which a function is defined and undefined. In this article, we explored the domain of the function $f(x) = \sqrt{x+6}$ and determined the correct interval notation.
References
- [1] "Functions" by Khan Academy
- [2] "Domain and Range" by Math Open Reference
- [3] "Square Root Function" by Wolfram MathWorld
Additional Resources
- [1] "Domain and Range" by IXL
- [2] "Functions" by Purplemath
- [3] "Square Root Function" by Mathway
Domain of a Function: Frequently Asked Questions =====================================================
Introduction
In our previous article, we explored the concept of the domain of a function and determined the domain of the function $f(x) = \sqrt{x+6}$. In this article, we'll 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 (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.
Q: How do I determine the domain of a function?
A: To determine the domain of a function, you need to consider the values of x for which the expression inside the function is defined. For example, if the function has a square root, you need to consider the values of x for which the expression inside the square root is non-negative.
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 (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.
Q: Can a function have a domain of all real numbers?
A: Yes, a function can have a domain of all real numbers. For example, the function $f(x) = x^2$ has a domain of all real numbers.
Q: Can a function have a domain of a single value?
A: Yes, a function can have a domain of a single value. For example, the function $f(x) = \frac{1}{x}$ has a domain of all real numbers except for 0.
Q: How do I write the domain of a function in interval notation?
A: To write the domain of a function in interval notation, you need to use the following notation:
-
(-\infty, a)$ to represent all real numbers less than a
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[a, \infty)$ to represent all real numbers greater than or equal to a
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(-\infty, a) \cup (b, \infty)$ to represent all real numbers less than a and greater than b
Q: Can a function have a domain that is a union of intervals?
A: Yes, a function can have a domain that is a union of intervals. For example, the function $f(x) = \frac{1}{x}$ has a domain of all real numbers except for 0, which can be written as $(-\infty, 0) \cup (0, \infty)$.
Q: How do I determine the domain of a function with a fraction?
A: To determine the domain of a function with a fraction, you need to consider the values of x for which the denominator is not equal to 0.
Q: Can a function have a domain that is a union of intervals with a single value?
A: Yes, a function can have a domain that is a union of intervals with a single value. For example, the function $f(x) = \frac{1}{x}$ has a domain of all real numbers except for 0, which can be written as $(-\infty, 0) \cup (0, \infty)$.
Conclusion
In conclusion, the domain of a function is the set of all possible input values (x-values) for which the function is defined. By understanding the concept of the domain, we can determine the values of x for which a function is defined and undefined. We hope this article has helped to answer some of the frequently asked questions about the domain of a function.
References
- [1] "Functions" by Khan Academy
- [2] "Domain and Range" by Math Open Reference
- [3] "Square Root Function" by Wolfram MathWorld
Additional Resources
- [1] "Domain and Range" by IXL
- [2] "Functions" by Purplemath
- [3] "Square Root Function" by Mathway