1. **State the problem:** We have a set of points $f = \{(a,2), (2,5), (-2,4), (a^2 + a, 5)\}$ and need to find values of $a$ such that $f$ is a function. A function assigns exactly one $y$-value to each $x$-value. 2. **Recall the function rule:** For $f$ to be a function, no two points can have the same $x$-coordinate with different $y$-values. 3. **Check given points:** - Points $(2,5)$ and $(-2,4)$ have distinct $x$-values. - Points $(a,2)$ and $(a^2 + a, 5)$ depend on $a$. 4. **Avoid $x$-value conflicts:** - $a$ must not equal $2$ or $-2$ to avoid conflict with existing points. - Also, $a$ and $a^2 + a$ must be distinct to avoid duplicate $x$-values. 5. **Set $a \neq 2$ and $a \neq -2$**. 6. **Check if $a = a^2 + a$:** $$a = a^2 + a \implies 0 = a^2 \implies a = 0$$ 7. **If $a=0$, then $a^2 + a = 0$, so points $(0,2)$ and $(0,5)$ have the same $x=0$ but different $y$-values, violating the function rule. So $a \neq 0$. 8. **Summary:** - $a \neq 0$, $a \neq 2$, $a \neq -2$. - For all other $a$, $f$ is a function. **Final answer:** $$a \in \mathbb{R} \setminus \{0, 2, -2\}$$