Arrange the following functions in a list so that each function is big-O of the next function. Label the functions from A to F in order, where A is the slowest growing and F is the fastest. log(x^220) / x 220 log(x) 220^220 x + (220 log(x)) 220^x - x^220 (x-220)!
Arrange the following functions in a list so that each function is big-O of the next function. Label the functions from A to F in order, where A is the slowest growing and F is the fastest. log(x^220) / x 220 log(x) 220^220 x + (220 log(x)) 220^x - x^220 (x-220)!
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Transcribed Image Text:### Big-O Complexity Ordering of Functions
**Task:** Arrange the following functions in a list so that each function is *big-O* of the next function. Label the functions from **A** to **F** in order, where **A** is the slowest growing and **F** is the fastest.
1. \( \frac{\log(x^{220})}{x} \)
2. \( 220 \log(x) \)
3. \( 220^{220} \)
4. \( x + (220 \log(x)) \)
5. \( 220^x - x^{220} \)
6. \( (x-220)! \)
**Big-O Notation Explanation:**
- **A** (Slowest growth): \( 220^{220} \)
- **B**: \( 220 \log(x) \)
- **C**: \( x + (220 \log(x)) \)
- **D**: \( \frac{\log(x^{220})}{x} \)
- **E**: \( 220^x - x^{220} \)
- **F** (Fastest growth): \( (x-220)! \)
**Explanation:**
1. \( 220^{220} \): A constant function.
2. \( 220 \log(x) \): A logarithmic function.
3. \( x + (220 \log(x)) \): A linear function plus a logarithmic term.
4. \( \frac{\log(x^{220})}{x} \): This simplifies to \( \frac{220 \log(x)}{x} \), which is smaller than \( x \).
5. \( 220^x - x^{220} \): An exponential function.
6. \( (x-220)! \): A factorial function, which grows faster than exponential functions.
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