∛x

Definition

The cube root of x, denoted as ∛x, is the value that, when cubed, gives the original number x. It is a radical function that represents the inverse of the cubic function, f(x) = x^3.

5 Must Know Facts For Your Next Test

1. The cube root function, ∛x, is the inverse of the cubic function, f(x) = x^3.
2. The domain of the cube root function is all real numbers, while the range is also all real numbers.
3. The graph of the cube root function is a continuous, increasing function that passes through the origin and has a characteristic S-shape.
4. The cube root function can be used to solve cubic equations by finding the value of x that satisfies the equation x^3 = a, where a is the given value.
5. The cube root function is one-to-one, meaning that each input value corresponds to a unique output value.

Review Questions

• Explain how the cube root function, ∛x, is the inverse of the cubic function, f(x) = x^3.
  • The cube root function, ∛x, is the inverse of the cubic function, f(x) = x^3, because it undoes the operation of cubing a number. Specifically, if f(x) = x^3, then the inverse function, ∛x, satisfies the equation ∛(x^3) = x. This means that the cube root function reverses the cubic function, taking the cube root of a number to find the original value that, when cubed, gives that number.
• Describe the key features of the graph of the cube root function, ∛x.
  • The graph of the cube root function, ∛x, has several notable features. First, it is a continuous, increasing function that passes through the origin, (0, 0). The graph has a characteristic S-shape, with the function growing more slowly near the origin and then increasing more rapidly as the input values become larger. Additionally, the cube root function has a domain and range that both include all real numbers, reflecting the fact that the cube root is defined for all real numbers, both positive and negative.
• Explain how the cube root function can be used to solve cubic equations of the form x^3 = a.
  • The cube root function, ∛x, can be used to solve cubic equations of the form x^3 = a, where a is a given value. To solve such an equation, one can simply take the cube root of both sides, resulting in the equation ∛(x^3) = ∛a, which simplifies to x = ∛a. This allows the original cubic equation to be solved by finding the cube root of the constant term, a. This process of using the inverse function to solve an equation is a key application of the cube root function in the context of inverse and radical functions.

Related terms

Radical Function: A radical function is a function that contains a radical expression, such as square roots, cube roots, or higher-order roots.

Inverse Function: An inverse function is a function that 'undoes' the operation of another function, reversing the input and output values.

Cubic Function: A cubic function is a polynomial function of degree 3, with the general form f(x) = ax^3 + bx^2 + cx + d.