1. **Problem 1: Find the value of 5c given the mapping:** Given the mapping: $$\begin{array}{ccccccc} x & -2 & -1 & 0 & 2 & 3 & 4 & \ldots & 5c \\ y & y & -1 & 1 & 5 & 7 & 9 & \ldots & 21 \\\end{array}$$ We see the values of $y$ for $x=2,3,4$ are $5,7,9$ respectively, and for $x=5c$, $y=21$. 2. **Step 1: Identify the pattern in $y$ values for given $x$ values.** From $x=2$ to $x=3$, $y$ increases from $5$ to $7$ (increase by 2). From $x=3$ to $x=4$, $y$ increases from $7$ to $9$ (increase by 2). This suggests $y$ increases by 2 for each increase of 1 in $x$. 3. **Step 2: Express $y$ as a function of $x$.** Since $y$ increases by 2 for each increase of 1 in $x$, the function is linear: $$y = 2x + b$$ 4. **Step 3: Find $b$ using a known point.** Use $x=2$, $y=5$: $$5 = 2(2) + b \implies 5 = 4 + b \implies b = 1$$ So, $$y = 2x + 1$$ 5. **Step 4: Use the function to find $5c$.** Given $y=21$ when $x=5c$: $$21 = 2(5c) + 1$$ $$21 - 1 = 2(5c)$$ $$20 = 2(5c)$$ $$\cancel{2}0 = \cancel{2}(5c)$$ $$10 = 5c$$ 6. **Step 5: Solve for $c$.** $$10 = 5c \implies c = \frac{10}{5} = 2$$ **Answer for Problem 1:** $c=2$ --- 2. **Problem 2: Find the image of $p$ in the mapping:** Given: $$\begin{array}{cccc} 1 & 2 & 3 & p \\ 3 & 5 & 7 & ? \\\end{array}$$ 3. **Step 1: Identify the pattern in the $y$ values.** The $y$ values are $3,5,7$ for $x=1,2,3$. The difference between consecutive $y$ values is $2$. 4. **Step 2: Express $y$ as a function of $x$.** Since $y$ increases by 2 for each increase of 1 in $x$, the function is: $$y = 2x + b$$ 5. **Step 3: Find $b$ using a known point.** Use $x=1$, $y=3$: $$3 = 2(1) + b \implies 3 = 2 + b \implies b = 1$$ So, $$y = 2x + 1$$ 6. **Step 4: Find the image of $p$.** $$y = 2p + 1$$ **Answer for Problem 2:** The image of $p$ is $2p + 1$ --- 3. **Problem 3: Find the image of $-4$ under the mapping $x \to \frac{1}{2}x - 2$.** 7. **Step 1: Substitute $x = -4$ into the function:** $$y = \frac{1}{2}(-4) - 2 = -2 - 2 = -4$$ **Answer for Problem 3:** The image of $-4$ is $-4$ --- 4. **Problem 4: Find the rule of the mapping given:** $$\begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ y & 0 & 1 & 4 & 9 & 16 \\\end{array}$$ 8. **Step 1: Identify the pattern in $y$.** $y$ values are $0,1,4,9,16$ which are perfect squares: $$0 = 0^2, 1 = 1^2, 4 = 2^2, 9 = 3^2, 16 = 4^2$$ 9. **Step 2: Write the rule:** $$y = x^2$$ **Answer for Problem 4:** The rule is $y = x^2$ --- 5. **Problem 5: Find the missing numbers $p$ and $q$ in the mapping:** $$\begin{array}{ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 \\ y & 3 & 5 & p & 9 & 11 & q \\\end{array}$$ 10. **Step 1: Identify the pattern in $y$ values.** Known $y$ values: $3,5,?,9,11,?$ Check differences: From $3$ to $5$: increase by $2$ From $5$ to $p$: unknown From $p$ to $9$: unknown From $9$ to $11$: increase by $2$ 11. **Step 2: Hypothesize the pattern.** Since $3$ to $5$ and $9$ to $11$ increase by $2$, assume $y$ increases by $2$ for each increase in $x$. 12. **Step 3: Find $p$ and $q$.** For $x=3$: $$p = 5 + 2 = 7$$ For $x=6$: $$q = 11 + 2 = 13$$ **Answer for Problem 5:** $p=7$, $q=13$