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

Question

Determine the continuity of the function f(x)=x3x f(x) = x^3 - x

Question 7

Answer

  • a. (0, 0)
  • b. (-∞, ∞)
  • c. (0, ∞)
  • d. (-∞, 0)
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Solution

To determine the continuity of the function f(x)=x3x f(x) = x^3 - x , we can follow these steps:

1. Break Down the Problem

We need to analyze the function f(x) 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)=x3x 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)=x3x 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) f(x) : The function is defined for all xR 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) f(x) is a polynomial, it is continuous for all x x .

Final Answer

The function f(x)=x3x f(x) = x^3 - x is continuous on the interval (,) (-\infty, \infty) . Therefore, the correct answer is: b. (-∞, ∞)

This problem has been solved

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