what are the domain and range of the function $f(x)=\frac{1}{4}(x^{3}+8x^{2}+5x - 4)$?

Explanation:

Step1: Recall domain rules for polynomials

Polynomial functions are defined for all real - valued inputs. The given function $f(x)=\frac{1}{4}(x^{3}+8x^{2}+5x - 4)$ is a polynomial function (cubic polynomial). So, the domain is all real numbers.

Step2: Analyze the end - behavior of the cubic polynomial

As $x\to+\infty$, $y = f(x)\to+\infty$ since the leading term of the polynomial $y=\frac{1}{4}x^{3}+2x^{2}+\frac{5}{4}x - 1$ is $\frac{1}{4}x^{3}$ and the coefficient of the leading term $\frac{1}{4}>0$. As $x\to-\infty$, $y = f(x)\to-\infty$. Since the function is continuous (all polynomials are continuous), the range is also all real numbers.

Answer:

Domain: $(-\infty,\infty)$; Range: $(-\infty,\infty)$