Verified Answer :

  • A cubic function is a polynomial where the highest degree term is \(x^3\).
  • Analyze the degree of each function: \(f(x) = 5x^3 + 2x + 1\) has degree 3, \(f(x) = 3x + 1\) has degree 1, \(f(x) = x^3 + x^4 + 5\) has degree 4, and \(f(x) = 3x^2 + x + 2\) has degree 2.
  • Only the function \(f(x) = 5x^3 + 2x + 1\) satisfies the definition of a cubic function.
  • The cubic function is \(\boxed{f(x)=5x^3+2x+1}\).

Explanation

  1. Definition of a Cubic Function and Given Data

A cubic function is defined as a polynomial of degree 3, which means the highest exponent of the variable \(x\) in the function is 3. The general form is \(f(x) = ax^3 + bx^2 + cx + d\), where \(a, b, c, d\) are constants and \(a eq 0\). We are given four functions to evaluate:

  1. \(f(x) = 5x^3 + 2x + 1\)
  2. \(f(x) = 3x + 1\)
  3. \(f(x) = x^3 + x^4 + 5\)
  4. \(f(x) = 3x^2 + x + 2\)
  1. Evaluating Each Function Degree

We examine the degree of each function by identifying the highest power of \(x\):

  • For \(f(x) = 5x^3 + 2x + 1\), the powers of \(x\) are 3, 1, and 0 (since 1 = \(1 \cdot x^0\)). The highest power is 3, making it a cubic function.
  • For \(f(x) = 3x + 1\), the highest power of \(x\) is 1. This is a linear function.
  • For \(f(x) = x^3 + x^4 + 5\), the powers of \(x\) are 3, 4, and 0. Rearranging gives \(f(x) = x^4 + x^3 + 5\). The highest power is 4, making it a quartic function.
  • For \(f(x) = 3x^2 + x + 2\), the highest power of \(x\) is 2. This is a quadratic function.
  1. Conclusion

Based on the evaluation, only \(f(x) = 5x^3 + 2x + 1\) has a degree of 3. Therefore, it is the only cubic function among the given choices. The final answer is \(\boxed{f(x)=5x^3+2x+1}\).

Examples

Cubic functions are widely used in modeling real-world phenomena. For example, if you are designing a storage box, the volume \(V\) of the box as a function of its side length \(x\) might be a cubic function, like \(V(x) = x^3\). Similarly, engineers might use cubic polynomials to describe the trajectory of a bridge's cable or to approximate smooth curves in computer-aided design software. Understanding the degree of a polynomial helps in selecting the appropriate mathematical model for these kinds of problems.

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