To determine the domain and range of the function \( f(x)=\frac{1}{4}(x^3 + 6x^2 + 5x - 4) \), we need to follow these steps:
1. Identify the Domain:
For any polynomial function, the domain is typically all real numbers. This means that \( f(x) \) can accept any real value of \( x \).
However, in this specific instance, we're given a limited domain of \( (-6.5, 2.5) \). This restriction means that we are only interested in values of \( x \) within the interval \( -6.5 \leq x \leq 2.5 \).
Therefore, the domain is:
[tex]\[
(-6.5, 2.5)
\][/tex]
2. Identify the Range:
The function \( f(x) \) is a cubic polynomial. Generally, the range of any odd-degree polynomial (such as \( x^3 \)) is all real numbers because as \( x \) approaches \( \infty \) or \( -\infty \), the function value will also approach \( \infty \) or \( -\infty \) respectively.
Despite the specific domain restriction in this case, the range of the function remains the same since it is bounded only by the properties of the polynomial itself.
Therefore, the range is:
[tex]\[
(-\infty, \infty)
\][/tex]
In summary, the domain and range of the function \( f(x)=\frac{1}{4}(x^3 + 6x^2 + 5x - 4) \) are:
- Domain: \( (-6.5, 2.5) \)
- Range: [tex]\( (-\infty, \infty) \)[/tex]
