To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=∛x? Is there any open interval over which the function is decreasing?

Verified step by step guidance

1. Recall that the function ƒ(x) = ∛x represents the cube root of x, which is the inverse of the cubic function y = x³.
2. Understand the shape of the graph of ƒ(x) = ∛x: it passes through the origin (0,0), is symmetric about the origin (odd function), and increases as x moves from negative to positive values.
3. Identify the graph by looking for a curve that starts from the bottom left (negative x and y), passes through the origin, and rises to the top right (positive x and y), showing a smooth, continuous increase.
4. To determine if there is any open interval where the function is decreasing, consider the derivative of ƒ(x) = ∛x, which is ƒ'(x) = \frac{1}{3x^{2/3}}; since this derivative is positive for all x ≠ 0, the function is increasing everywhere except possibly at x = 0.
5. Conclude that there is no open interval where ƒ(x) = ∛x is decreasing because its derivative does not become negative on any interval.

Key Concepts

Cube Root Function

The cube root function, denoted as ƒ(x) = ∛x, returns the number that, when cubed, equals x. Its graph is symmetric about the origin and passes through (0,0). Unlike square roots, it is defined for all real numbers, including negatives, producing an S-shaped curve.

Graph Interpretation

Interpreting graphs involves recognizing key features such as shape, intercepts, and symmetry. For ƒ(x) = ∛x, the graph increases continuously without breaks, reflecting the function's domain and range. Identifying the correct graph requires matching these characteristics.

Increasing and Decreasing Intervals

A function is increasing on an interval if its output values rise as x increases, and decreasing if outputs fall. For ƒ(x) = ∛x, the function is strictly increasing over its entire domain, meaning there are no intervals where it decreases.