The function $f(x)$ is defined as follows:

• $f(x) = 3x - 4$ for $3 \leq x \leq 4$
• $f(x) = x^3 - 4$ for $4 < x \leq 5$

Let's calculate $f(0)$, $f(1)$, $f(4)$, and $f(5)$:

1. $f(0)$: Since $0$ is less than $3$, and there's no definition provided for $f(x)$ when $x < 3$, $f(0)$ is undefined.
2. $f(1)$: Similarly, $1$ is also less than $3$, so $f(1)$ is also undefined.
3. $f(4)$: For $x = 4$, we use the function $f(x) = 3x - 4$ because $x$ exactly equals $4$ and this interval is inclusive. Thus: $f(4) = 3(4) - 4 = 12 - 4 = 8$
4. $f(5)$: For $x = 5$, we use the function $f(x) = x^3 - 4$ because $x$ falls within the interval $4 < x \leq 5$. Thus: $f(5) = 5^3 - 4 = 125 - 4 = 121$

Summary:

• $f(0)$ is undefined
• $f(1)$ is undefined
• $f(4) = 8$
• $f(5) = 121$

Do you have any other questions or need further details on this problem? Here are some follow-up topics we can explore:

1. What would $f(x)$ be if $x = 3$?
2. How would $f(x)$ change if $x$ were slightly above $4$, like $4.01$?
3. Can we graph the function $f(x)$ to visually understand the values across different intervals?
4. How does the discontinuity between the piecewise segments at $x = 4$ affect the function graphically?
5. What's the general approach to solving piecewise functions like this one?

Tip: When dealing with piecewise functions, always pay close attention to the intervals and their boundaries to determine which expression to use.