Question

which is a cubic function?
○ $f(x) = 3x + 1$
○ $f(x) = 3x^2 + x + 2$
○ $f(x) = 5x^3 + 2x + 1$
○ $f(x) = x^3 + x^4 + 5$

Explanation:

Step1: Recall the definition of a cubic function

A cubic function is a polynomial function of degree 3, which means the highest power of \(x\) in the function is 3, and the function is of the form \(f(x)=ax^{3}+bx^{2}+cx + d\) where \(a
eq0\).

Step2: Analyze each option

• Option 1: \(f(x)=3x + 1\). The highest power of \(x\) is 1, so it is a linear function (degree 1), not cubic.
• Option 2: \(f(x)=3x^{2}+x + 2\). The highest power of \(x\) is 2, so it is a quadratic function (degree 2), not cubic.
• Option 3: \(f(x)=5x^{3}+2x + 1\). The highest power of \(x\) is 3, and the coefficient of \(x^{3}\) (\(5\)) is non - zero. So this is a cubic function.
• Option 4: \(f(x)=x^{3}+x^{4}+5\). The highest power of \(x\) is 4 (from the \(x^{4}\) term), so it is a quartic function (degree 4), not cubic.

Answer:

\(f(x) = 5x^{3}+2x + 1\) (the third option: \(f(x)=5x^{3}+2x + 1\))