For which values of x is the function ​f(x)equals=3 x cubed+x+1 ​continuous?

The function \( f(x) = 3x^3 + x + 1 \) is a polynomial function. Polynomial functions are continuous for all real values of \( x \). This is because polynomials are built from basic operations (addition, subtraction, multiplication, and exponentiation) that are continuous everywhere.

Thus, the function \( f(x) = 3x^3 + x + 1 \) is continuous for **all real values of \( x \)**, i.e., \( x \in (-\infty, \infty) \).

Would you like further details on the continuity of polynomial functions?

Here are 5 related questions to further explore this concept:
1. What makes a function discontinuous?
2. How do you determine if a rational function is continuous?
3. Are all polynomial functions continuous over their entire domain?
4. How does the derivative of a polynomial behave with respect to continuity?
5. What are some real-world applications of continuous polynomial functions?

**Tip:** A function is continuous if you can draw its graph without lifting your pen from the paper!

For which values of x is the function f(x) = 3x^3 + x + 1 continuous?

Explore the continuity of the polynomial function f(x) = 3x^3 + x + 1. This problem demonstrates that polynomial functions are continuous for all real values of x, covering key concepts in calculus and continuity. Suitable for students in Grades 11-12.

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