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Radical Functions

Radical Functions

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Understanding Radical Functions and Expressions

In our previous lessons, we have explored various polynomial functions such as f(x) = x2 and f(x) = x3, where the base number is represented by the x variable and the exponent (or power) is indicated by the superscript of 2 or 3. Now, let's delve into the inverses of these functions - the square root and cube root, also known as radicals.

The term "radical" is derived from the Latin word "radix" meaning "root". In English, the word "root" refers to the source or origin of something. Similarly, in mathematics, the root of a number is its source or origin. For example, when we square the number 3, the result is 9. Here, 3 is the root and 9 is the square of that root.

Radical Expressions and Functions

A radical expression is an expression containing the radical symbol '√', also known as a radical. Examples include √4 and ∛27, where the numbers within the radical symbol are called the radicand.

The most commonly known radicals are the square root and cube root, represented by the symbols √ and ∛ respectively.

Solving for Square Roots and Cube Roots

Solving expressions with a square root is a straightforward process - we simply take the square root of the radicand. The square root of a number is the number that, when multiplied by itself, yields the given radicand. Similarly, the cube root of a number is the number that, when multiplied by itself three times, equals the given radicand.

Evaluating Radical Functions

Evaluating radical functions follows the same method as solving regular functions. We just need to substitute the given value of x into the function to find the value of f(x). For example:

  • Given the radical function f(x) = √x, evaluate f(9).
  • Given the radical function g(x) = ∛x, evaluate g(64).

Graphs and Transformations of Radical Functions

As mentioned before, the radical functions y = √x and y = ∛x are the inverse functions of the polynomial functions y = x2 and y = x3, respectively. In this section, we will compare their graph shapes, transformations, and study their domains and ranges.

Domain and Range of Radical Functions

To identify the domain and range of a radical function, let's first review the characteristics of the square root and cube root functions.

The Square Root Function

The square root function is represented by √x and its graph resembles a half parabola, as shown below:

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