Question

for parts (a) and (b) below, guess what might be latifah’s rule for the function. in each case, if n is the input and l(n) is the answer, express l(n) in terms of n.
a. a possible rule could be l(n) =

n | 6 | 8 | 9 | 12
l(n) | 14 | 20 | 23 | 32

b. a possible rule could be l(n) =

n | 3 | 5 | 7 | 12
l(n) | 14 | 30 | 54 | 149

Explanation:

Part a: Analyze the first table

We have the following pairs \((n, L(n))\): \((6, 14)\), \((8, 20)\), \((9, 23)\), \((12, 32)\). Let's check the relationship between \(n\) and \(L(n)\).

Let's assume a linear relationship \(L(n)=an + b\). For \(n = 6\), \(L(6)=14\), so \(6a + b=14\). For \(n = 8\), \(L(8)=20\), so \(8a + b=20\). Subtract the first equation from the second: \((8a + b)-(6a + b)=20 - 14\), which gives \(2a=6\), so \(a = 3\). Then substitute \(a = 3\) into \(6a + b=14\): \(18 + b=14\), so \(b=-4\). Let's check for \(n = 9\): \(3\times9-4=27 - 4=23\) (which matches). For \(n = 12\): \(3\times12-4=36 - 4=32\) (which matches). So the rule for part a is \(L(n)=3n - 4\).

Part b: Analyze the second table

We have the pairs \((n, L(n))\): \((3, 14)\), \((5, 30)\), \((7, 54)\), \((12, 149)\). Let's check the relationship. Let's try \(L(n)=n^{2}+5\) for \(n = 3\): \(3^{2}+5=9 + 5=14\) (matches). For \(n = 5\): \(5^{2}+5=25 + 5=30\) (matches). For \(n = 7\): \(7^{2}+5=49 + 5=54\) (matches). For \(n = 12\): \(12^{2}+5=144 + 5=149\) (matches). So the rule for part b is \(L(n)=n^{2}+5\).

Answer:

a. \(L(n)=3n - 4\)
b. \(L(n)=n^{2}+5\)