1. The problem is to determine if the given relation \( f \) from set \( D = \{1, 2, 3, 4\} \) to set \( Y = \{a, b, c, d\} \) defined by: \( f(1) = a, f(2) = b, f(3) = b, f(4) = c \) is a function. 2. By definition, a relation \( f: D \to Y \) is a function if every element in the domain \( D \) maps to exactly one element in the codomain \( Y \). 3. Here, each element in \( D \) has a unique output value: - \( 1 \to a \) - \( 2 \to b \) - \( 3 \to b \) - \( 4 \to c \) There are no elements in \( D \) with multiple outputs. Thus, \( f \) is a function. 4. The co-domain is the set \( Y = \{a, b, c, d\} \). 5. The range (or image) is the subset of the co-domain actually mapped to by \( f \). Here, the range is \( \{a, b, c\} \) because these are the elements that appear as outputs. Final answers: - \( f \) is a function. - Co-domain: \( \{a, b, c, d\} \). - Range: \( \{a, b, c\} \).