To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=x³? What is its range?

Verified step by step guidance

1. Recall the general shape of the graph of the function \(f(x) = x^3\). It is a cubic function with an S-shaped curve that passes through the origin (0,0). The graph increases without bound as \(x\) becomes large positive and decreases without bound as \(x\) becomes large negative.
2. Compare the given graphs to identify the one that matches this S-shaped cubic curve. The correct graph will have symmetry about the origin and will not be a parabola or linear function.
3. Once the correct graph is identified, consider the range of the function \(f(x) = x^3\). Since cubic functions are continuous and increase or decrease without bound, the output values cover all real numbers.
4. Express the range of \(f(x) = x^3\) using interval notation. Because the function can take any real value, the range is all real numbers, which is written as \((-\infty, \infty)\).
5. Summarize that the graph of \(f(x) = x^3\) is the cubic S-shaped curve passing through the origin, and its range is all real numbers, \((-\infty, \infty)\).

Key Concepts

Cubic Function and Its Graph

A cubic function is a polynomial of degree three, typically written as f(x) = x³. Its graph is an S-shaped curve that passes through the origin, increasing from negative infinity to positive infinity. Understanding the shape helps identify the correct graph among options.

Range of a Function

The range of a function is the set of all possible output values (f(x)) it can produce. For f(x) = x³, since the cubic function extends infinitely in both positive and negative directions, its range is all real numbers, denoted as (-∞, ∞).

Interpreting Graphs of Functions

Interpreting graphs involves recognizing key features like intercepts, end behavior, and symmetry. For f(x) = x³, the graph passes through (0,0), is symmetric about the origin (odd function), and shows continuous growth, which helps distinguish it from other basic graphs.