∛x

Definition

The cube root of x, denoted as ∛x, is the value that, when multiplied by itself three times, equals x. It is a radical function that is used to represent 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, and its range is also all real numbers.
3. The graph of the cube root function is an S-shaped curve that passes through the origin and is increasing for all real numbers.
4. The cube root function is continuous and differentiable for all real numbers.
5. The cube root function is used to solve equations of the form x^3 = a, where a is a real number.

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 if we have a cubic equation of the form f(x) = x^3, then the cube root function, ∛x, can be used to find the value of x that satisfies the equation. In other words, if f(x) = y, then the inverse function, ∛y, will give the value of x that makes the equation true.
• Describe the key properties of the cube root function, ∛x, and how they relate to its inverse relationship with the cubic function.
  • The cube root function, ∛x, has several important properties that relate to its inverse relationship with the cubic function, f(x) = x^3. First, the domain and range of the cube root function are both the set of all real numbers, which means it can be used to solve any cubic equation. Second, the graph of the cube root function is an S-shaped curve that is continuous and differentiable, reflecting the smooth, continuous nature of the inverse relationship. Finally, the cube root function can be used to solve equations of the form x^3 = a, where a is a real number, by applying the cube root to both sides of the equation.
• Analyze how the cube root function, ∛x, can be used to transform and simplify expressions involving cubic functions.
  • The cube root function, ∛x, can be used to transform and simplify expressions involving cubic functions in a variety of ways. For example, if we have an expression of the form x^3 + 3x^2 - 5x + 2, we can use the cube root function to isolate the variable x by taking the cube root of both sides of the equation. This allows us to rewrite the expression in a simpler form that is easier to work with and understand. Additionally, the cube root function can be used to simplify more complex expressions involving cubic functions by breaking them down into their component parts and applying the inverse relationship between the cube root function and the cubic function.

Related terms

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

Inverse Function: An inverse function is a function that 'undoes' the operation of another function, so that if f(x) = y, then the inverse function f^(-1)(y) = x.

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