Function Domain Range Gap
Suppose we define $ \ell(n) $ as follows: If $ n $ is an integer from $ 0 $ to $ 20, $ inclusive, then $ \ell(n) $ is the number of letters in the English spelling of the number $ n; $ otherwise, $ \ell(n) $ is undefined. For example, $ \ell(11)=6, $ because "eleven" has six letters, but $ \ell(23) $ is undefined, because $ 23 $ is not an integer from $ 0 $ to $ 20 $.
How many numbers are in the domain of $ \ell(n) $ but not the range of $ \ell(n)?$
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
We can make a table showing the values of $ \ell(n): $ $$\begin{array}{c | c | c || c | c | c || c | c | c}
n & \text{spelling} & \ell(n) & n & \text{spelling} & \ell(n) & n & \text{spelling} & \ell(n) \\
\hline
0 & \text{zero} & 4 & 7 & \text{seven} & 5 & 14 & \text{fourteen} & 8 \\
1 & \text{one} & 3 & 8 & \text{eight} & 5 & 15 & \text{fifteen} & 7 \\
2 & \text{two} & 3 & 9 & \text{nine} & 4 & 16 & \text{sixteen} & 7 \\
3 & \text{three} & 5 & 10 & \text{ten} & 3 & 17 & \text{seventeen} & 9 \\
4 & \text{four} & 4 & 11 & \text{eleven} & 6 & 18 & \text{eighteen} & 8 \\
5 & \text{five} & 4 & 12 & \text{twelve} & 6 & 19 & \text{nineteen} & 8 \\
6 & \text{six} & 3 & 13 & \text{thirteen} & 8 & 20 & \text{twenty} & 6
\end{array}$$ Thus, $ \ell(n) $ can take every integer value from $ 3 $ to $ 9 $. The numbers that are in the domain of $ \ell(n) $ but not the range are $$0,1,2,10,11,12,13,14,15,16,17,18,19,20,$$ and there are $ \boxed{14} $ numbers in this list.
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