The Square Root Function: Definition, Properties, and Applications in Mathematics

Square Root Function

The square root function is a mathematical function that maps a non-negative real number to its principal square root

The square root function is a mathematical function that maps a non-negative real number to its principal square root. It is denoted by the symbol √x or x^(1/2).

The square root of a number x, denoted as √x, is a value that, when multiplied by itself, gives the original number x. In other words, if y = √x, then y^2 = x.

The square root function is defined for non-negative real numbers, as the square root of a negative number results in a complex number. For example, the square root of -9 is not a real number.

Graphically, the square root function is represented as a curve that starts at the origin (0,0) and increases gradually as x increases. The graph of the square root function is only valid for non-negative values of x, as shown below.

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x-axis
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The domain of the square root function is all non-negative real numbers, [0, ∞). This means that any non-negative value can be input into the function. For example, sqrt(4) = 2, sqrt(9) = 3, and sqrt(0) = 0.

The range of the square root function is also all non-negative real numbers, [0, ∞). This means that any non-negative value can be output by the function. For example, sqrt(4) = 2, sqrt(9) = 3, and sqrt(0) = 0.

The square root function has several important properties:

1. Square root of a product: sqrt(ab) = sqrt(a) * sqrt(b). This means that the square root of a product is equal to the product of the square roots of each factor.

2. Square root of a quotient: sqrt(a/b) = sqrt(a) / sqrt(b). This means that the square root of a quotient is equal to the quotient of the square roots of each term.

3. Square root of a power: sqrt(a^b) = (sqrt(a))^b. This means that the square root of a power is equal to the power of the square root of the base.

4. Square root of a square: sqrt(a^2) = |a|. This means that the square root of a squared number is equal to the absolute value of the original number.

When working with the square root function, it is important to consider the context and any restrictions on the values of x. For example, when solving equations involving the square root function, we often need to consider the positive and negative square roots as both may be valid solutions.

I hope this explanation helps you understand the square root function! If you have any further questions or need clarification, please feel free to ask.

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