1. The problem is to determine if a function is surjective (onto). 2. A function $f: A \to B$ is surjective if for every element $b$ in the codomain $B$, there exists at least one element $a$ in the domain $A$ such that $f(a) = b$. 3. The key formula or condition is: $$\forall b \in B, \exists a \in A \text{ such that } f(a) = b.$$ This means the function covers the entire codomain. 4. To check surjectivity: - Identify the codomain $B$. - For an arbitrary element $b$ in $B$, try to solve the equation $f(a) = b$ for $a$. - If you can find such an $a$ for every $b$, the function is surjective. 5. Important rules: - Surjectivity depends on the codomain, not just the range. - If the function is from real numbers to real numbers, check if the equation $f(a) = b$ has a solution for all real $b$. 6. Example: For $f(x) = 2x + 3$, to check surjectivity: - Let $y$ be any real number. - Solve $2x + 3 = y$ for $x$. - $$x = \frac{y - 3}{2}$$ - Since $x$ exists for every real $y$, $f$ is surjective. 7. Summary: To tell if a function is surjective, verify that for every element in the codomain, there is a preimage in the domain that maps to it.