Exploring the Cube Root Function: Definition, Domain, Range, Graph, Examples, Properties, and Operations

Cube Root Function

The cube root function, denoted as f(x) = ∛x, is a mathematical function that calculates the cube root of a given number

The cube root function, denoted as f(x) = ∛x, is a mathematical function that calculates the cube root of a given number. It is the inverse function of raising a number to the power of 3 (cubing).

To understand the cube root function, let’s break down the key components:

1. Definition: The cube root of a number x is a number y such that y^3 = x. In other words, if we raise y to the power of 3, we will obtain x.

2. Domain and Range: The domain of the cube root function is all real numbers since we can take the cube root of any real number. The range is also all real numbers, as there is a cube root for every real number.

3. Graph: The graph of the cube root function is a curve that starts at negative infinity, passes through the origin (0, 0), and extends to positive infinity. The shape of the curve is similar to that of a square root function, but it has a steeper incline.

4. Examples:
– ∛8 = 2 since 2^3 = 8
– ∛(-27) = -3 since (-3)^3 = -27
– ∛0 = 0 since 0^3 = 0

5. Properties:
– The cube root of a negative number is negative because the cube of a negative number is negative.
– The cube root of a positive number is positive because the cube of a positive number is positive.
– The cube root of zero is zero.
– The cube root function is an odd function, which means that f(-x) = -f(x). This is because it reflects across the y-axis.

6. Operations:
– Addition and subtraction: ∛(a ± b) ≠ ∛a ± ∛b. The cube root function doesn’t have the same distributive property as addition and subtraction, so you can’t simply add or subtract the cube roots of two numbers individually.
– Multiplication: ∛(ab) = ∛a ∙ ∛b. You can multiply the cube roots of two numbers separately to find the cube root of their product.
– Division: ∛(a/b) = ∛a / ∛b. You can divide the cube roots of two numbers separately to find the cube root of their quotient.

It’s essential to remember that the cube root function is distinct from the square root function (√x), which calculates the square root of a number. The cube root function is specifically used to find the value that, when cubed, equals a given number.

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