1. Let's start by stating the problem: We want to understand when a function $a$ is one-to-one (injective). 2. A function $f$ is one-to-one if and only if for every pair of inputs $x_1$ and $x_2$, whenever $f(x_1) = f(x_2)$, it implies that $x_1 = x_2$. 3. This means no two different inputs map to the same output. 4. To check if a function $a$ is one-to-one, we can use the horizontal line test on its graph: if any horizontal line intersects the graph more than once, the function is not one-to-one. 5. Alternatively, algebraically, solve $a(x_1) = a(x_2)$ and check if it implies $x_1 = x_2$. 6. For example, if $a(x) = 2x + 3$, then $a(x_1) = a(x_2)$ means $2x_1 + 3 = 2x_2 + 3$, which simplifies to $2x_1 = 2x_2$, so $x_1 = x_2$. Hence, $a$ is one-to-one. 7. Important rule: Linear functions with nonzero slope are always one-to-one. 8. Quadratic functions like $a(x) = x^2$ are not one-to-one over all real numbers because $a(1) = a(-1) = 1$ but $1 \neq -1$. 9. To summarize, a function $a$ is one-to-one if it never assigns the same output to two different inputs. Final answer: A function $a$ is one-to-one if $a(x_1) = a(x_2)$ implies $x_1 = x_2$ for all $x_1, x_2$ in its domain.