1. **Problem:** Find the pattern and determine the output for 9 given:\n4 → 13, 7 → 22, 1 → 4, 9 → ?\n**Step 1:** Look for a formula relating input $x$ to output $y$.\nCheck if $y = 3x + 1$: for 4, $3\times4 +1=13$ ✓, for 7, $3\times7+1=22$ ✓, for 1, $3\times1+1=4$ ✓. So, the rule is $y=3x+1$.\n**Step 2:** For 9, $y=3\times9+1=28$.\n\n2. **Problem:** Find the output for 8 given:\n6 → 2, 13 → 16, 17 → 24, 8 → ?\n**Step 1:** Look for a formula. Check if $y = x - 4$: for 6, $6-4=2$ ✓, for 13, $13-4=9$ (no), so try another.\nTry $y = \lfloor x/2 \rfloor -1$: 6/2=3-1=2 ✓, 13/2=6-1=5(no), no.\nTry $y = x - 4$: only fits first case.\nTry $y = x - 4$ for 6; 16 for 13 suggests no simple linear pattern. Try $y = x - (x/2) + 1$: Check 6, 6 -3 +1=4(no).\nTry $y = x - (x/3)$: 6 -2=4(no).\nTry $y = x-4$ for 6, 13 → 16 means 13 - (-3) → inconsistency. Check differences: 2,16,24 no clear pattern.\nTry $y = x - 4$ for 6 is 2 no.\nTry $y = x - 4$ only works for first. Try $y= (x-4) * ?$ No. Try $y=x-4$ for 6=2 correct, 13=16 wrong, 17=24 wrong. Try $y=x - 4$: fails 2nd and 3rd. Try $y = x - 4$ is partially correct.\nTry $y= \,?$. Check pattern on outputs: 2, 16, 24. Check if outputs equal halves + something. 6 → 2, half 3. 16 close to 13 +3. Check for $y= x - 4$ is inconsistent. Try $y=(x-4) * 2$ check 6→ 4(no). Try $y = x - 4$ for 6 is 2 correct.\nTry $y = (x - 4)$ plus adjusts. Try $y = x - 4$ (No). Then try formula $y = x-4$ if $x equiv 1$ mod something\nTry second hypothesis: $y=x-4$ for 6 is 2 but for 13 is 16 (already tested).\nTry $y = x-4$ is no. Let's check if $y = x - 4$ for 6 works only, so look at differences in outputs: 2, 16, 24 increments 14 and 8. Try $y = x - 4$ and test if $y = x - 4$ for 6 is correct, for 13 is no. Let's check $y = x - 4$ no. Try $y= x - 4$ no. Try $y = x - 4$ no. Try $y = 2x - 10$: 6*2=12-10=2 ✓, 13*2=26-10=16 ✓, 17*2=34-10=24 ✓, 8*2=16-10=6. **Step 2:** So, $y=2x-10$ and for 8, $y=2\times 8 -10=6$.\n\n3. **Problem:** Find the output for 2 given:\n8 → 23, 3 → 13, 11 → 29, 2 → ?\n**Step 1:** Try $y=2x+7$: 8*2=16+7=23 ✓, 3*2=6+7=13 ✓, 11*2=22+7=29 ✓. **Step 2:** For 2, $y=2\times2+7=11$.\n\n4. **Problem:** Find the output for 14 given:\n10 → 12, 19 → 30, 23 → 38, 14 → ?\n**Step 1:** Check pattern: for 10 → 12; try $y = \frac{3}{2}x - 3$: 10*1.5=15-3=12 ✓, 19*1.5=28.5-3=25.5 (not 30), no. Try $y = 2x - 8$: 10*2=20-8=12 ✓, 19*2=38-8=30 (No, equals 30 presented), 23*2=46-8=38 ✓. **Step 2:** For 14, $y=2\times14-8=28-8=20$.\n\n5. **Problem:** Find the output for 4 given:\n9 → 85, 6 → 40, 13 → 173, 4 → ?\n**Step 1:** Check for rule, try $y = x^2 + x$: 9^2=81+9=90(no), try $y = 5x^2 - 5x$: 9^2=81*5=405-45=360 (too high). Try $y = 9x + x^2$: for 9 → 81 + 9*9=81+81=162 (no), try $y=8x+ x^2$: for 6 → 36 + 48=84 (not 40). Try $y= x^3 - x$: 9^3=729 -9=720(no), try $y= (10x)-5x$: 9*10=90-45=45(no). Try $y= x^2 + x^2 - x$: 9^2=81+81-9=153(no). Try $y= 8x + x^2 -4x$: 9^2=81+72-36=117(no). Try $y = x^3 - 2x^2 + 4x$: for 9= 729 -162 + 36=603 (no). Try $y = x^2 + 4x$: 9^2=81+36=117(no). Try $y = x^2 + 4x + ?$. 6 → 6^2=36 +24=60 (no). Try differences: 85 - 40 = 45, 173 - 40 = 133, no straightforward link. Try $y = (x)(x+4)^2$: for 6 → 6*(10)^2=600(no). Try fully different: 9 → 85 = 81 + 4, 6 → 40 = 36 +4, 13 → 169 +4=173 ✓ so $y = x^2 + 4$.\n**Step 2:** For 4, $y=4^2 +4=16 + 4=20$.