1. **Problem:** Classify the types of the given relations and identify if given relations are functions. 2. **Relation a)** $x \to \frac{x}{2}$ with pairs $(2,4), (4,16), (6,36)$. - Check if each input $x$ maps to exactly one output. - Here, $2 \to 4$, $4 \to 16$, $6 \to 36$. - Each input has one output, so it is a function. 3. **Relation b)** $x \to \sqrt{x}$ with pairs $(4,3), (9,2), (9,-2), (9,-3)$. - Input $9$ maps to multiple outputs $2, -2, -3$. - A function must have exactly one output per input. - So, this is not a function. 4. **Relation c)** $x \to x^2$ with pairs $(3,9), (2,4), (-3,9)$. - Each input maps to exactly one output. - So, this is a function. 5. **Relation d)** Type of number with pairs $(4, \text{Prime}), (9, \text{Even}), (-3, \text{Even})$. - The classification is incorrect (4 is not prime, 9 and -3 are not even). - Also, this is a relation but not a function because the classification is inconsistent. 6. **Function check a)** Sets $A = \{p,q,r\}$ and $B = \{1,2,3\}$ with mappings $p \to 1$, $q \to 2$, $r \to 3$. - Each element in $A$ maps to exactly one element in $B$. - This is a function. 7. **Function check b)** Sets $A = \{p,q,r\}$ and $B = \{a,b,c,d\}$ with mappings $p \to a$, $q \to b$, $r \to c$, and $p \to d$. - Element $p$ maps to two outputs $a$ and $d$. - This is not a function. 8. **Function check c)** Sets $A = \{a,b,c\}$ and $B = \{d,a,r\}$ with mappings $a \to d$, $b \to a$, $c \to r$. - Each element in $A$ maps to exactly one element in $B$. - This is a function. **Summary:** - Relation a) is a function. - Relation b) is not a function. - Relation c) is a function. - Relation d) is not a function. - Function check a) is a function. - Function check b) is not a function. - Function check c) is a function.