1. The problem is to understand and apply the concept of functions, including definitions, types, domain, range, and related properties. 2. A function $f$ from a set $A$ to a set $B$ is a rule that assigns each element in $A$ exactly one element in $B$. The set $A$ is called the domain, $B$ the codomain, and the set of all actual outputs is the range. 3. Important types of functions: - Into function: Not all elements of $B$ are mapped to by elements of $A$. - Onto (surjective) function: Every element of $B$ is mapped to by at least one element of $A$. - One-one (injective) function: Different elements of $A$ map to different elements of $B$. - Bijective function: Both one-one and onto; a perfect pairing between $A$ and $B$. 4. To determine the value of a function for given inputs, substitute the input into the function's formula and simplify. 5. Example: Let $f(x) = 2x + 3$. Find $f(4)$. 6. Substitute $x=4$: $$f(4) = 2(4) + 3 = 8 + 3 = 11$$ 7. So, the value of the function at $x=4$ is 11. This process applies to any function and input values. Final answer: For $f(x) = 2x + 3$, $f(4) = 11$.