To find the domain and range of the function [tex]\(f(x)=\frac{1}{4}\left(x^3 + 6x^2 + 5x - 4\right)\)[/tex], we can follow the steps below:
### Domain:
1. Determine the domain of the function.
2. A cubic polynomial (like [tex]\(x^3 + 6x^2 + 5x - 4\)[/tex]) is defined for all real numbers because there is no division by zero or square roots of negative numbers involved in the expression.
3. So, the basic domain of a cubic polynomial would normally be all real numbers, [tex]\((-\infty, \infty)\)[/tex].
However, considering the specifics given, the domain of the function [tex]\(f(x)\)[/tex] is restricted to the interval [tex]\((-8.5, 2.5)\)[/tex]. This might be due to practical constraints or the specific context in which the function is used.
Therefore, the domain of the function is:
[tex]\[ \boxed{(-8.5, 2.5)} \][/tex]
### Range:
1. Analyze the range of the function.
2. For cubic functions, the output can cover all real numbers. This is because a cubic function can produce very large positive and negative values, as well as cover every value in between.
3. Hence, typically, the range of a cubic polynomial would be [tex]\((-\infty, \infty)\)[/tex].
Given the context, it is clear that the function is indeed capable of producing every real value. Hence, the range of [tex]\(f(x)\)[/tex] is:
[tex]\[ \boxed{(-\infty, \infty)} \][/tex]
### Summary:
The domain and range of the function [tex]\(f(x)=\frac{1}{4}\left(x^3 + 6x^2 + 5x - 4\right)\)[/tex] are:
- Domain: [tex]\( (-8.5, 2.5) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
