Understanding the Square Root Function: Properties, Graph, and Solving Equations

Square Root Function

The square root function, denoted as √x or √(x), is a mathematical function that returns the positive square root of a given number x

The square root function, denoted as √x or √(x), is a mathematical function that returns the positive square root of a given number x. In other words, it is the inverse of the square function.

To understand the square root function, let’s consider an example. If we have the equation √x = 4, how do we solve for x?

To isolate x, we need to square both sides of the equation to remove the square root symbol:

(√x)² = (4)²

This simplifies to:

x = 16

So, in this case, the value of x that satisfies the equation √x = 4 is x = 16.

Now, let’s look at some properties of the square root function:

1. Domain: The square root function is defined for non-negative real numbers. The domain of the function is x ≥ 0.

2. Range: The range of the square root function is y ≥ 0. It only gives non-negative outputs.

3. Graph: The graph of the square root function is a curve that starts at the origin (0,0) and extends to the right, gradually increasing as x increases. The graph is symmetric with respect to the y-axis.

4. Even though the square root function is the inverse of the square function, it is important to note that √x has a single value for positive values of x. For example, √4 = 2, not ±2. However, for negative values of x, the square root is not defined in the real number system (there is no real number whose square is negative).

5. The square root function can be used to solve quadratic equations by setting the equation equal to zero and finding the values of x that satisfy it.

In summary, the square root function is a mathematical function that returns the positive square root of a given number. It has domain x ≥ 0, range y ≥ 0, and its graph is a curve starting at the origin. It is the inverse of the square function.

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