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Chapter 1: Problem 121

Determine whether the statement is true or false. Justify your answer. A function with a square root cannot have a domain that is the set of real numbers.

Short Answer

Expert verified
The statement is false. The function \(f(x) = \sqrt{x^2}\) is a counter-example that demonstrates a function with a square root can have a domain that is the set of all real numbers.

Step by step solution

01

Understand the Domain of the Function and Square Root Functions

The domain of a function consists of all the numbers that the function can take as an input. In the case of square root functions, the input is located under the square root sign. For a square root function to be defined, the input under the square root sign must be a non-negative number (i.e., greater than or equal to zero) because the square root of a negative number is not a real number.
02

Evaluate the Statement

The statement 'A function with a square root cannot have a domain that is the set of real numbers' is suggesting that there can be no square root function that can take all real numbers as its domain. To evaluate the validity of this statement, consider a square root function where the input under the square root sign is always non-negative, regardless of the real number that is inputted into the function. The function \(f(x) = \sqrt{x^2}\) is an example of such a function. For this function, the input under the square root sign is the square of a real number, which is always non-negative. Therefore, the domain of this function is all real numbers.
03

Conclude the Correctness of the Statement

Since the function \(f(x) = \sqrt{x^2}\) is a square root function that can have a domain that is the set of all real numbers, the statement 'A function with a square root cannot have a domain that is the set of real numbers' is not true. Therefore, the statement is false.

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Most popular questions from this chapter

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