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Chapter 1: Problem 93

A function with a square root cannot have a domain that is the set of real numbers.

Short Answer

Expert verified
Correct. A function with a square root doesn't have a domain that is all real numbers. The domain is limited to nonnegative real numbers.

Step by step solution

01

Understand the Function's Structure

A function involving a square root would have the general form \(f(x) = \sqrt{x}\). The function is defined only when the expression under the square root (or 'radicand') is a positive number or zero.
02

Define the Domain

The set of permissible input values for \(x\) (the function's domain) is determined by the condition that the radicand must be greater than or equal to zero. Thus for the function \(f(x) = \sqrt{x}\), the domain is \(x \geq 0\). This means that the input \(x\) can be any real number, as long as it is not less than zero.
03

Answer the Task

While some real numbers are valid inputs for the function, it is not the case that all real numbers are allowed. Specifically, negative numbers are not within the function's domain. Hence, a function with a square root does not have as its domain the set of all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
A square root function often appears in mathematical expressions as something like \(f(x) = \sqrt{x}\). This type of function involves finding the square root of a number. The square root of any given number is what, when multiplied by itself, gives the original number. For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
  • Square root functions are unique because they only work with non-negative numbers.
  • The result of a square root is always a non-negative number as well.

When dealing with square root functions, remember only to include values for which the square root is real, meaning no negative numbers are under the root. This condition directly affects the domain of the function.
Real Numbers Domain
The domain of a function refers to all possible input values that the function can accept. For functions involving square roots, the domain isn't just any real number. Instead, it's limited based on what makes sense for the operation.
  • A real number is any number that exists on the number line, including both positive and negative numbers, as well as fractions and decimals.
  • However, for a function like \(f(x) = \sqrt{x}\), the domain is specifically restricted to values where \(x \geq 0\).

This limitation arises because the square root of a negative number isn't a real number. Therefore, the domain of a square root function is not all real numbers, but just non-negative ones (0 and any positive number).
Radicand
In the context of a square root function, the radicand is the term or expression under the square root symbol. For example, in \(\sqrt{x}\), \(x\) is the radicand. Determining the radicand is crucial because it dictates the restrictions on the domain of the function.
  • The radicand must be greater than or equal to zero for the square root to return a real number.
  • Negative radicands aren't within the domain of square root functions because they would result in imaginary numbers, not real ones.

Understanding the concept of a radicand helps you determine which values of \(x\) are permissible in your function, shaping the real number domain effectively.

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Most popular questions from this chapter

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