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Observing the functions f(x)=4x+3

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Observing the Functions

Observing the Functions:

Functions:

  • f(x)=4x+3f(x) = 4x + 3 (Linear function)
  • g(x)=(53)xg(x) = \left( \frac{5}{3} \right)^x (Exponential function)

Sentences to Complete:

  1. Over the interval ______\_\_\_\_\_\_______, the average rate of change of gg is greater than the average rate of change of ff.
  2. As the value of xx increases, the average rates of change of ff and gg ___________, respectively.
  3. When the value of xx is equal to 7, the value of _______________\_\_\_\_\_\_\_\_\_\_\_\_\_\_\________________.
  4. It can be further generalized that a quantity increasing exponentially will ___________ exceed a quantity increasing linearly.



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1 Answer

    • Over the interval [3, 6], the average rate of change of gg is greater than the average rate of change of ff.
    • As the value of xx increases, the average rates of change of ff and gg remain constant and increase, respectively.
    • When the value of xx is equal to 7, the value of g(x)g(x) will exceed the value of f(x)f(x).
    • It can be further generalized that a quantity increasing exponentially will eventually exceed a quantity increasing linearly.