Determine the continuity of the function 𝑓𝑥=𝑥3-𝑥Question 7Answera.(0, 0)b.(-∞, ∞)c.(0, ∞)d.(-∞, 0)

To determine the continuity of the function $f(x) = x^3 - x$, we can follow these steps:

1. Break Down the Problem

We need to analyze the function $f(x)$ to check if it is continuous over its entire domain or particular intervals. A function is continuous if it is defined and does not have any breaks, holes, or asymptotes.

2. Relevant Concepts

A polynomial function is continuous everywhere on the real number line. The function $f(x) = x^3 - x$ is a polynomial function, so we can conclude its continuity based on this property.

3. Analysis and Detail

The function $f(x) = x^3 - x$ is a polynomial of degree three. Since all polynomial functions are continuous across their entire domain, we can assert:

• Domain of $f(x)$: The function is defined for all $x \in \mathbb{R}$ (from $-\infty$ to $+\infty$).
• Continuity: There are no breaks or discontinuities in the function since it is a polynomial.

4. Verify and Summarize

Since $f(x)$ is a polynomial, it is continuous for all $x$.

Final Answer

The function $f(x) = x^3 - x$ is continuous on the interval $(-\infty, \infty)$. Therefore, the correct answer is: b. (-∞, ∞)