1. The problem is to determine if a function $f$ is injective (one-to-one). 2. A function $f$ is injective if and only if for every $x_1$ and $x_2$ in the domain, whenever $f(x_1) = f(x_2)$, it implies that $x_1 = x_2$. 3. The formula or condition to check injectivity is: $$\text{If } f(x_1) = f(x_2) \Rightarrow x_1 = x_2$$ 4. Important rules: - If two different inputs produce the same output, the function is not injective. - To test injectivity, assume $f(x_1) = f(x_2)$ and try to prove $x_1 = x_2$. 5. Example: For $f(x) = 2x + 3$, assume $f(x_1) = f(x_2)$: $$2x_1 + 3 = 2x_2 + 3$$ 6. Subtract 3 from both sides: $$2x_1 + \cancel{3} = 2x_2 + \cancel{3}$$ $$2x_1 = 2x_2$$ 7. Divide both sides by 2: $$\frac{\cancel{2}x_1}{\cancel{2}} = \frac{\cancel{2}x_2}{\cancel{2}}$$ $$x_1 = x_2$$ 8. Since $x_1 = x_2$ follows from $f(x_1) = f(x_2)$, the function $f(x) = 2x + 3$ is injective. In summary, to know if $f$ is injective, check if $f(x_1) = f(x_2)$ implies $x_1 = x_2$ for all $x_1, x_2$ in the domain.