√x

Definition

The square root of x is a mathematical operation that finds the positive value that, when multiplied by itself, equals the original number x. It represents the inverse operation of squaring a number and is a fundamental function in algebra and calculus.

5 Must Know Facts For Your Next Test

1. The square root function is denoted as $\sqrt{x}$ or $x^{1/2}$, where $x$ is the input variable.
2. The square root function is an example of a radical function, which is a function that contains a root (such as square root, cube root, etc.) of a variable.
3. The square root function is the inverse of the squaring function, meaning that $\sqrt{x^2} = x$ and $x^2 = \sqrt{x}$.
4. The domain of the square root function is the set of non-negative real numbers, as the square root of a negative number is not defined in the real number system.
5. The range of the square root function is the set of non-negative real numbers, as the square root of a number can never be negative.

Review Questions

• Explain the relationship between the square root function and the squaring function.
  • The square root function, $\sqrt{x}$, is the inverse of the squaring function, $x^2$. This means that the square root function 'undoes' the squaring operation, and vice versa. Specifically, if $y = \sqrt{x}$, then $x = y^2$, and if $y = x^2$, then $x = \sqrt{y}$. The square root function is a radical function that represents the positive value that, when multiplied by itself, equals the original number $x$.
• Describe the domain and range of the square root function.
  • The domain of the square root function, $\sqrt{x}$, is the set of non-negative real numbers, as the square root of a negative number is not defined in the real number system. The range of the square root function is also the set of non-negative real numbers, as the square root of a number can never be negative. This is because the square root function represents the positive value that, when multiplied by itself, equals the original number $x$.
• Explain how the properties of the square root function relate to the concepts of inverse functions and radical functions.
  • The square root function, $\sqrt{x}$, is an example of an inverse function. As the inverse of the squaring function, $x^2$, the square root function 'undoes' the squaring operation. This relationship between the square root function and the squaring function is a key property of inverse functions. Additionally, the square root function is a radical function, which is a function that contains a root (such as square root, cube root, etc.) of a variable. The properties of the square root function, including its domain, range, and inverse relationship with the squaring function, are important characteristics of radical functions in general.

Related terms

Radical Function: A radical function is a function that contains a square root, cube root, or other root of a variable. These functions have unique properties and behaviors compared to polynomial functions.

Inverse Function: An inverse function is a function that 'undoes' the operation of another function. The square root function is the inverse of the squaring function, as it reverses the squaring operation.

Domain and Range: The domain of a square root function is the set of non-negative real numbers, as the square root of a negative number is not defined in the real number system. The range of a square root function is the set of non-negative real numbers.