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How does g(t)=9tg(t)=9^t change over the interval from t=1t=1 to t=3t=3?\newlineChoices: \newline[g(t) decreases by a factor of 81]\left[\text{g(t) decreases by a factor of } 81\right]\newline[g(t) increases by a factor of 81]\left[\text{g(t) increases by a factor of } 81\right]\newline[g(t) increases by 18%]\left[\text{g(t) increases by } 18\%\right]\newline[g(t) decreases by 9%]\left[\text{g(t) decreases by } 9\%\right]

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Q. How does g(t)=9tg(t)=9^t change over the interval from t=1t=1 to t=3t=3?\newlineChoices: \newline[g(t) decreases by a factor of 81]\left[\text{g(t) decreases by a factor of } 81\right]\newline[g(t) increases by a factor of 81]\left[\text{g(t) increases by a factor of } 81\right]\newline[g(t) increases by 18%]\left[\text{g(t) increases by } 18\%\right]\newline[g(t) decreases by 9%]\left[\text{g(t) decreases by } 9\%\right]
  1. Find g(1)g(1): We have the function g(t)=9tg(t) = 9^t. Find the value of g(1)g(1).\newlineSubstitute t=1t = 1 into g(t)=9tg(t) = 9^t.\newlineg(1)=91g(1) = 9^1\newlineg(1)=9g(1) = 9
  2. Find g(3)g(3): We have the function g(t)=9tg(t) = 9^t. Find the value of g(3)g(3).\newlineSubstitute t=3t = 3 into g(t)=9tg(t) = 9^t.\newlineg(3)=93g(3) = 9^3\newlineg(3)=729g(3) = 729
  3. Calculate Factor Change: We found:\newlineg(1)=9g(1) = 9\newlineg(3)=729g(3) = 729\newlineCalculate the factor by which g(t)g(t) has changed from t=1t = 1 to t=3t = 3.\newlineDivide g(3)g(3) by g(1)g(1).\newlineFactor change = g(3)g(1)=7299\frac{g(3)}{g(1)} = \frac{729}{9}\newlineFactor change = 8181
  4. Determine Increase or Decrease: Determine if g(t)g(t) increases or decreases.\newlineSince 729729 is greater than 99, g(t)g(t) increases from t=1t = 1 to t=3t = 3.
  5. Summary: We found:\newlineFactor change: 8181\newlineBehavior of g(t)g(t): increases\newlineHow does g(t)=9tg(t) = 9^t change from t=1t = 1 to t=3t = 3?\newlineWe found that g(t)g(t) increases and the factor by which it increases is 8181.\newlineg(t)g(t) increases by a factor of 8181.

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