1. The problem states that $j(x) = k(m(x))$ and provides values for $m(x)$, $k(x)$, and $j(x)$ at certain points. 2. We are given the following values: - For $x = 7,8,9,10,11$: - $m(x) = 11,10,9,8,7$ - $k(x) = 7,11,10,\text{blank},\text{blank}$ - $j(x) = 10,8,9,11,\text{blank}$ - $X = 11,10,9,8,7$ 3. Since $j(x) = k(m(x))$, to find $j(x)$ for each $x$, we first find $m(x)$, then find $k$ at that value. 4. Let's fill in the missing values step-by-step: - For $x=7$: - $m(7) = 11$ - $k(11)$ is blank, so we need to find $k(11)$. - From $j(7) = k(m(7)) = k(11) = 10$ (given), so $k(11) = 10$. - For $x=8$: - $m(8) = 10$ - $k(10) = 11$ (given) - $j(8) = k(10) = 11$ but given $j(8) = 8$, so this is inconsistent. However, the problem states $j(8) = 8$, so we must check carefully. Actually, the problem states $j(8) = 8$, but $k(10) = 11$ from the table. Since $j(8) = k(m(8)) = k(10)$, this suggests $j(8) = 11$, but given $j(8) = 8$. This is a contradiction, so likely the transcription is off or the problem expects us to fill missing $k(x)$ values. 5. Let's try to fill missing $k(x)$ values for $x=8$ and $x=7$: - For $k(8)$ and $k(7)$, values are missing. 6. Using $j(x) = k(m(x))$, and given $j(8) = 8$, $m(8) = 10$, so $k(10) = 8$. But $k(10)$ is given as 11, so this is inconsistent. 7. Let's check the given data carefully: - $m(x)$ row: 11, 10, 9, 8, 7 for $x=7,8,9,10,11$ - $k(x)$ row: 7, 11, 10, blank, blank for $x=7,8,9,10,11$ - $j(x)$ row: 10, 8, 9, 11, blank for $x=7,8,9,10,11$ 8. Since $j(x) = k(m(x))$, for $x=7$: - $j(7) = k(m(7)) = k(11)$ - $j(7) = 10$ (given) - So $k(11) = 10$ 9. For $x=8$: - $j(8) = k(m(8)) = k(10)$ - $j(8) = 8$ (given) - So $k(10) = 8$ 10. For $x=9$: - $j(9) = k(m(9)) = k(9)$ - $j(9) = 9$ (given) - So $k(9) = 9$ 11. For $x=10$: - $j(10) = k(m(10)) = k(8)$ - $j(10) = 11$ (given) - So $k(8) = 11$ 12. For $x=11$: - $j(11) = k(m(11)) = k(7)$ - $j(11)$ is blank - $k(7) = 7$ (given) 13. Now we can fill the missing $k(x)$ values: - $k(11) = 10$ - $k(10) = 8$ - $k(9) = 9$ - $k(8) = 11$ - $k(7) = 7$ 14. The completed $k(x)$ row is: 7, 11, 9, 8, 10 for $x=7,8,9,10,11$. 15. The completed $j(x)$ row is: 10, 8, 9, 11, 7 for $x=7,8,9,10,11$. 16. The $m(x)$ row remains as given: 11, 10, 9, 8, 7. Summary: | x | 7 | 8 | 9 | 10 | 11 | |----|---|---|---|----|----| | m(x) | 11 | 10 | 9 | 8 | 7 | | k(x) | 7 | 11 | 9 | 8 | 10 | | j(x) | 10 | 8 | 9 | 11 | 7 | This satisfies $j(x) = k(m(x))$ for all $x$. Final answer: - $k(11) = 10$ - $k(10) = 8$ - $k(9) = 9$ - $k(8) = 11$ - $k(7) = 7$ - $j(11) = 7$