Which statement is true about the function f(x) = √x?
A. The domain of the graph is all real numbers.
B. The range of the graph is all real numbers.
C. The domain of the graph is all real numbers less than or equal to 0.
D. The range of the graph is all real numbers less than or equal to 0.
Answer (2)
The function is f ( x ) = x , which can be written as f ( x ) = x 4 1 .
The domain is determined by the values of x for which the function is defined, which is x ≥ 0 .
The range is determined by the possible output values of the function, which is f ( x ) ≥ 0 .
Therefore, the domain is all real numbers greater than or equal to 0, and the range is all real numbers greater than or equal to 0.
Explanation
Understanding the Function We are given the function f ( x ) = x . We need to determine the domain and range of this function and identify the correct statement from the given options.
Determining the Domain The domain of a function is the set of all possible input values (x) for which the function is defined. Since we have a square root function, the expression inside the square root must be greater than or equal to 0. In this case, we have two square roots, so we need to ensure that both x and x are greater than or equal to 0. Thus, x ≥ 0 . Therefore, the domain of the function is all real numbers greater than or equal to 0.
Determining the Range The range of a function is the set of all possible output values (f(x)) that the function can produce. Since x ≥ 0 , the square root of x, x , will also be greater than or equal to 0. Taking the square root again, x , will also result in a non-negative value. Thus, f ( x ) ≥ 0 . Therefore, the range of the function is all real numbers greater than or equal to 0.
Comparing with Given Statements Now, let's compare our findings to the given statements:
The domain of the graph is all real numbers. (Incorrect, as x must be greater than or equal to 0)
The range of the graph is all real numbers. (Incorrect, as f ( x ) must be greater than or equal to 0)
The domain of the graph is all real numbers less than or equal to 0. (Incorrect, as x must be greater than or equal to 0)
The range of the graph is all real numbers less than or equal to 0. (Incorrect, as f ( x ) must be greater than or equal to 0)
However, we found that the domain is all real numbers greater than or equal to 0, and the range is all real numbers greater than or equal to 0. Among the options, none of them is fully correct. However, if we consider the correct statements:
The domain of the graph is all real numbers greater than or equal to 0. The range of the graph is all real numbers greater than or equal to 0.
Conclusion Based on our analysis, the domain of f ( x ) = x is x ≥ 0 , and the range is f ( x ) ≥ 0 . Therefore, the correct answer is:
The domain of the graph is all real numbers greater than or equal to 0. The range of the graph is all real numbers greater than or equal to 0.
Examples
Understanding the domain and range of functions like f ( x ) = x is crucial in many real-world applications. For example, when modeling physical quantities that cannot be negative, such as length or time, we need to ensure that our function's domain and range reflect these constraints. Imagine you're designing a garden and using a function to determine the area you can cover based on the amount of fencing you have. The domain would represent the possible lengths of fencing (which can't be negative), and the range would represent the possible areas you can enclose (also non-negative).
The function f ( x ) = x has a domain of all real numbers greater than or equal to 0, and a range of all real numbers greater than or equal to 0. Therefore, none of the provided statements (A, B, C, or D) are true regarding the function. The correctly stated domain and range are both non-negative real numbers.
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