Input-Output Rules - SHSAT Math

1. Based on the table, which rule best represents $y$ in terms of $x$?
   Answer: $y = 2^{x}$ Outputs double each time the input increases by 1: $2^{1}=2,\;2^{2}=4,\;2^{3}=8,\;2^{4}=16$. B gives 3→9 not 8. C produces 3→10. D gives fractions for small inputs.
2. Refer to the table to answer the question. Which rule describes the relationship between $x$ and $y$?
   Answer: $y = 3x - 4$ For each input, multiply by 3 and subtract 4: $3(3)-4=5,\;3(5)-4=11,\;3(7)-4=17,\;3(9)-4=23$. Only choice A produces all listed outputs. B gives 3→5 but 5→9. C gives 3→5 but 5→21. D gives non-integer outputs for most inputs.
3. Use the values in the table to answer the question. Which rule generates every output shown?
   Answer: $y=|x-1|$ Taking the absolute value after subtracting 1 works: $|0-1|=1,\;|1-1|=0,\;|2-1|=1,\;|3-1|=2\,$. A gives negative for 0. C gives 0→1 but 1→2. D gives 0→−1.
4. Use the table to find the input–output rule.
   Answer: $y = x^{2} + 1$ Square each input and add 1: $1^{2}+1=2,\;2^{2}+1=5,\;3^{2}+1=10,\;4^{2}+1=17$. Only $y=x^{2}+1$ matches. B produces 1→3. A gives 1→0. D gives 1→1.
5. Use the table to identify the rule that matches every input–output pair.
   Answer: $y = 2x + 1$ Each input is doubled and then 1 is added: $2(2)+1=5,\;2(4)+1=9,\;2(6)+1=13,\;2(8)+1=17$. Only $y=2x+1$ fits all four pairs. B gives 2→5 but fails for 4→9 (it gives 7). C gives 2→5 but 4→11. D gives 2→5 and 4→17, which is wrong for 4.
6. Use the table to answer the question. Which rule matches every pair in the table?
   Answer: $y = x^{2}+2x$ Compute $x^{2}+2x$: $1^{2}+2(1)=3,\;2^{2}+2(2)=8,\;3^{2}+2(3)=15,\;4^{2}+2(4)=24$, matching the table. A gives 1→9 not 3. B gives 1→5 not 3. D gives 1→8 not 3.
7. Use the table to determine which rule generates the outputs.
   Answer: $y = x( x^{2} + 1 )$ Compute $x(x^{2}+1)=x^{3}+x$: 1→2, 2→10, 3→30, 4→68, matching. A gives 4→36. B gives 3→24. D gives 4→44.
8. Look at the table to answer the question. Which rule correctly produces the given outputs?
   Answer: $y = \dfrac{6}{x}$ Dividing 6 by the input: 1→6, 2→3, 3→2, 6→1. A gives 1→1. B gives 2→9. D gives negative values.
9. Look at the table to answer the question. Which rule correctly links the inputs and outputs?
   Answer: $y = \dfrac{x^{2}}{4} + 2$ Compute $\dfrac{x^{2}}{4}+2$: for x=0, 4, 8, 12 gives 2,6,18,38. A yields 0,4,16,36. B yields bigger numbers. C is linear.
10. Based on the table, which equation represents the relationship between $x$ and $y$?
    Answer: $y = x^{2} - x$ Compute $x^{2}-x$: $2^{2}-2=2,\;3^{2}-3=6,\;4^{2}-4=12,\;5^{2}-5=20$, matching the table. B gives 2→7 not 2. C gives 2→6 not 2. D gives 2→12 not 2.
11. Look at the table to answer the question. Which rule produces the outputs from the given inputs?
    Answer: $y = ( -2)^{x}$ Powers of −2 alternate sign and double magnitude: $(-2)^{0}=1, (-2)^{1}=-2, (-2)^{2}=4, (-2)^{3}=-8$ matching 1,−2,4,−8. A gives 0→2. B gives 0→−2. D gives 2→0.
12. Using the table, choose the rule that matches the data.
    Answer: $y = 3x^{2} - 2$ Compute $3x^{2}-2$: 1→1, 2→10, 3→25, 4→46, matching. A gives 1→1 but 2→7. B gives 2→7. D is linear.
13. Use the table to identify the rule.
    Answer: $y = x - 2(\text{integer part of }\frac{x}{2})$ The outputs alternate 1,0,1,0 showing the remainder when x is divided by 2. For odd x, remainder is 1; for even x, remainder is 0. A gives pattern 1,-1,1,-1 (negative values). C doubles inputs giving 2,4,6,8. D subtracts 2 yielding -1,0,1,2.
14. Refer to the table to answer the question. Identify the rule that produces the listed outputs.
    Answer: $y = \dfrac{x}{2} - 1$ Halve each input and subtract 1: $\tfrac{4}{2}-1=1,\;\tfrac{8}{2}-1=3,\;\tfrac{12}{2}-1=5,\;\tfrac{16}{2}-1=7$. B gives 4→7 not 1. C gives 4→1.5 not 1. D gives 4→1 correctly but 8→5 not 3.
15. Using the table, select the rule that fits all data pairs.
    Answer: $y = (x - 3)^{2}$ Test each rule against the data pairs in the table: (3, 0), (4, 1), (5, 4) and (6, 9). For $y = (x - 3)^{2}$: $(0)^2 = 0$, $(1)^2 = 1$, $(2)^2 = 4$, $(3)^2 = 9$. Every pair fits. For $y = x^{2} - 9$: $x = 3$ gives 0, which fits, but $x = 4$ gives 7, not 1. Checking only the first row makes this rule look right. For $y = x^{2} - 3$: $x = 3$ gives 6, not 0. For $y = x^{2} - 6x + 6$: $x = 3$ gives $9 - 18 + 6 = -3$, not 0; this is the expanded square with the constant term wrong. The only rule that fits all four pairs is $y = (x - 3)^{2}$.
16. Use the table to select the correct input–output rule.
    Answer: $y = 5 - x$ Subtract the input from 5: $5-(-1)=6,\;5-0=5,\;5-1=4,\;5-2=3$. A gives -1→12. B gives -1→4. C gives outputs of ±5.
17. Refer to the table to answer the question. What rule maps $x$ to $y$?
    Answer: $y = \sqrt{x} + 1$ Square roots: $\sqrt{0}+1=1,\;\sqrt{1}+1=2,\;\sqrt{4}+1=3,\;\sqrt{9}+1=4$. B gives 1→2 but 4→17. C gives 0→1 but 1→√2. D gives 4→4 not 3.
18. The table shown below lists some values of a function $r$. Which rule is consistent with all the values in the table shown?
    Answer: $r(x) = \dfrac{(x-1)^2}{4}$ For $r(x) = \frac{(x-1)^2}{4}$: $r(1) = 0$ ✓, $r(3) = 4/4 = 1$ ✓, $r(5) = 16/4 = 4$ ✓, $r(7) = 36/4 = 9$ ✓, $r(9) = 64/4 = 16$ ✓. Choice A gives $1, 2, 3, 4, 5$. Choice B gives $0, 2, 6, 12, 20$ — matches $x=1$ only. Choice C gives $0, 1, 2, 3, 4$ — matches $x=1, 3$ only.
19. Refer to the table to determine the rule for $y$.
    Answer: $y=x^{3}$ Cube each input: $1^{3}=1,\;2^{3}=8,\;3^{3}=27,\;4^{3}=64$. B gives 3→9. C gives 4→10. D gives 3→13.

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