What Is The Domain Of The Function $f(x) = \sqrt{(x-1)^2}$?(Enter 'r' For All Real Numbers.)
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 $f(x) = \sqrt{(x-1)^2}$.
Understanding the Function
The given function is $f(x) = \sqrt{(x-1)^2}$. This function involves the square root of a squared expression. To understand the domain of this function, we need to consider the properties of the square root and squared expressions.
Properties of Square Root and Squared Expressions
The square root of a number is defined only for non-negative real numbers. In other words, if $a \geq 0$, then $\sqrt{a}$ is defined. On the other hand, the square of a number is always non-negative, i.e., $(x-1)^2 \geq 0$ for all real numbers x.
Analyzing the Function
Now, let's analyze the function $f(x) = \sqrt{(x-1)^2}$. Since the expression inside the square root is always non-negative, the function is defined for all real numbers x. However, we need to consider the case when the expression inside the square root is equal to zero.
Case 1: Expression Inside the Square Root is Equal to Zero
When the expression inside the square root is equal to zero, the function is also equal to zero. In other words, if $(x-1)^2 = 0$, then $f(x) = 0$. This occurs when x = 1.
Case 2: Expression Inside the Square Root is Greater than Zero
When the expression inside the square root is greater than zero, the function is also greater than zero. In other words, if $(x-1)^2 > 0$, then $f(x) > 0$. This occurs when x ≠1.
Conclusion
Based on the analysis, we can conclude that the domain of the function $f(x) = \sqrt{(x-1)^2}$ is all real numbers. In other words, the domain of the function is $(-\infty, \infty)$. This is because the expression inside the square root is always non-negative, and the function is defined for all real numbers x.
Final Answer
The final answer is: r
Additional Information
The domain of a function is an essential concept in mathematics, and it's crucial to understand it when dealing with functions. In this article, we explored the domain of the function $f(x) = \sqrt{(x-1)^2}$. We analyzed the properties of the square root and squared expressions and concluded that the domain of the function is all real numbers.
Related Topics
- Domain of a function
- Square root and squared expressions
- Properties of functions
References
- [1] "Functions" by Khan Academy
- [2] "Domain of a function" by Math Open Reference
- [3] "Square root and squared expressions" by Wolfram MathWorld
Introduction
In our previous article, we explored the domain of the function $f(x) = \sqrt{(x-1)^2}$. We concluded that the domain of the function is all real numbers. In this article, we'll answer some frequently asked questions related to the domain of the function.
Q&A
Q1: What is the domain of the function $f(x) = \sqrt{(x-1)^2}$?
A1: The domain of the function $f(x) = \sqrt{(x-1)^2}$ is all real numbers. In other words, the domain of the function is $(-\infty, \infty)$.
Q2: Why is the domain of the function all real numbers?
A2: The domain of the function is all real numbers because the expression inside the square root is always non-negative. In other words, $(x-1)^2 \geq 0$ for all real numbers x.
Q3: What happens when the expression inside the square root is equal to zero?
A3: When the expression inside the square root is equal to zero, the function is also equal to zero. In other words, if $(x-1)^2 = 0$, then $f(x) = 0$. This occurs when x = 1.
Q4: What happens when the expression inside the square root is greater than zero?
A4: When the expression inside the square root is greater than zero, the function is also greater than zero. In other words, if $(x-1)^2 > 0$, then $f(x) > 0$. This occurs when x ≠1.
Q5: Is the domain of the function $f(x) = \sqrt{(x-1)^2}$ the same as the domain of the function $f(x) = (x-1)^2$?
A5: No, the domain of the function $f(x) = \sqrt{(x-1)^2}$ is not the same as the domain of the function $f(x) = (x-1)^2$. The domain of the function $f(x) = (x-1)^2$ is also all real numbers, but the function $f(x) = (x-1)^2$ is always non-negative, whereas the function $f(x) = \sqrt{(x-1)^2}$ is equal to zero when x = 1.
Q6: Can the domain of the function $f(x) = \sqrt{(x-1)^2}$ be changed?
A6: No, the domain of the function $f(x) = \sqrt{(x-1)^2}$ cannot be changed. The domain of a function is a fixed set of values that the function can accept without resulting in an undefined or imaginary output.
Q7: How does the domain of the function $f(x) = \sqrt{(x-1)^2}$ relate to the range of the function?
A7: The domain of the function $f(x) = \sqrt{(x-1)^2}$ is all real numbers, whereas the range of the function is also all real numbers, but the function is always non-negative.
Conclusion
In this article, we answered some frequently asked questions related to the domain of the function $f(x) = \sqrt{(x-1)^2}$. We concluded that the domain of the function is all real numbers and that the expression inside the square root is always non-negative.
Final Answer
The final answer is: r
Additional Information
The domain of a function is an essential concept in mathematics, and it's crucial to understand it when dealing with functions. In this article, we explored the domain of the function $f(x) = \sqrt{(x-1)^2}$ and answered some frequently asked questions related to the domain of the function.
Related Topics
- Domain of a function
- Square root and squared expressions
- Properties of functions
References
- [1] "Functions" by Khan Academy
- [2] "Domain of a function" by Math Open Reference
- [3] "Square root and squared expressions" by Wolfram MathWorld