Math Problem Statement

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

Solution

The function f(x)=3x3+x+1f(x) = 3x^3 + x + 1 is a polynomial function. Polynomial functions are continuous for all real values of xx. This is because polynomials are built from basic operations (addition, subtraction, multiplication, and exponentiation) that are continuous everywhere.

Thus, the function f(x)=3x3+x+1f(x) = 3x^3 + x + 1 is continuous for all real values of xx, i.e., x(,)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!

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Math Problem Analysis

Mathematical Concepts

Calculus
Continuity
Polynomial Functions

Formulas

f(x) = 3x^3 + x + 1

Theorems

Continuity of Polynomial Functions

Suitable Grade Level

Grades 11-12