Math Problem Statement

For which values of x is the function f(x) = x^3 + x + 3 continuous?

Solution

To solve this problem, we need to determine the continuity of the function f(x)=x3+x+3f(x) = x^3 + x + 3.

Step-by-Step Solution:

  1. Nature of the Function:
    The given function f(x)=x3+x+3f(x) = x^3 + x + 3 is a polynomial. Polynomial functions are continuous everywhere on the real line.

  2. Polynomial Functions:
    For any polynomial function, there are no breaks, jumps, or points of discontinuity. The function is smooth and continuous across all real numbers.

  3. Conclusion:
    Since f(x)f(x) is a polynomial, it is continuous for all real values of xx. Therefore, the correct answer is:

    B. It is continuous for all xx.


Would you like more details or have any questions about this solution?

Here are 5 related questions for further exploration:

  1. What are the key properties that make a polynomial function continuous?
  2. How do you test the continuity of more complex, non-polynomial functions?
  3. What is the difference between continuity and differentiability in a function?
  4. Can a function be continuous but not differentiable? If so, how?
  5. What is the formal definition of continuity at a point?

Tip: When dealing with functions, remember that all polynomials are continuous and differentiable everywhere on their domain, which is the set of all real numbers.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Continuity

Formulas

f(x) = x^3 + x + 3

Theorems

Polynomials are continuous for all real numbers.

Suitable Grade Level

Grades 9-12