1. **State the problem:** We are given the function $F(x) = \frac{1}{x}$ defined on the interval $0 < x < 4$. We need to determine whether this function is bounded on this interval. 2. **Recall the definition of boundedness:** A function $f(x)$ is bounded on an interval if there exists a real number $M$ such that for all $x$ in the interval, $|f(x)| \leq M$. 3. **Analyze the function:** The function $F(x) = \frac{1}{x}$ is defined for $x$ in $(0,4)$. As $x$ approaches $0$ from the right, $\frac{1}{x}$ becomes very large (tends to $+\infty$). As $x$ approaches $4$, $\frac{1}{x}$ approaches $\frac{1}{4} = 0.25$. 4. **Check the behavior near the endpoints:** - Near $x=0^+$, $F(x) \to +\infty$, so the function is not bounded above. - Near $x=4$, $F(x)$ is finite and equals $0.25$. 5. **Conclusion:** Since $F(x)$ grows without bound as $x$ approaches $0$ from the right, the function is **not bounded** on the interval $(0,4)$. **Final answer:** The function $F(x) = \frac{1}{x}$ is not bounded on the interval $0 < x < 4$ because it tends to infinity as $x$ approaches $0$ from the right.