1. **State the problem:** We are given values of a function $f$ at $x = -5$ and $x = 4$, with $f(-5) = -2$ and $f(4) = 3$. The functions $f$ and $g$ satisfy $f(g(x)) = x$ for all real $x$. We need to find which point must lie on the graph of $g$. 2. **Recall the property of inverse functions:** Since $f(g(x)) = x$, $g$ is the inverse function of $f$. This means that if $f(a) = b$, then $g(b) = a$. 3. **Apply the property to given values:** From the table, $f(-5) = -2$ implies $g(-2) = -5$. 4. Similarly, $f(4) = 3$ implies $g(3) = 4$. 5. **Check the options:** - A: $(-5, -2)$ means $g(-5) = -2$, which is not true. - B: $(\frac{1}{3}, \frac{1}{4})$ no information supports this. - C: $(3, 4)$ means $g(3) = 4$, which matches our result. - D: $(5, 2)$ no information supports this. 6. **Conclusion:** The point $(3,4)$ must be on the graph of $g$. **Final answer:** $(3,4)$