Complexity examples

Fig 1

• Red: x2
• Blue: x

Fig 2a

• Red: 0.0001 x2
• Blue: x

Fig 2b

• Red: 0.0001 x2
• Blue: x

The above figures demonstrate that the form of the function (x2 versus x) is the important factor, as x becomes very large.

Fig 3

• Red: 0.0001 x2
• Blue: x
• Green: 3x

The above figure demonstrates that the form of the function (x2 versus x) is more important than the constant (0.0001, 1, and 3 that mulitplies the function, as x becomes very large.

Fig 4

• Red: 0.0001 x2
• Blue: x
• Green: 100 log x
• Magenta: Sum of the above three function

The above figure demonstrates that as x becomes large, the dominant term (0.0001 x2), in a sum of terms, contributes an increasingly larger fraction to the sum.