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Listen Determine whether the function represents exponential growth or exponential decay. f(t)=(1)/(2)((3)/(2))t The function represents Identify the percent rate of change.

Listen\newlineDetermine whether the function represents exponential growth or exponential decay.\newlinef(t)=12(32)t f(t)=\frac{1}{2}\left(\frac{3}{2}\right) t \newlineThe function represents\newlineIdentify the percent rate of change.

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Q. Listen\newlineDetermine whether the function represents exponential growth or exponential decay.\newlinef(t)=12(32)t f(t)=\frac{1}{2}\left(\frac{3}{2}\right) t \newlineThe function represents\newlineIdentify the percent rate of change.
  1. Analyze base for growth: We need to analyze the base of the exponential function to determine if it represents growth or decay.\newlineThe base of the exponential function is (32)(\frac{3}{2}), which is greater than 11.\newlineSince the base is greater than 11, the function represents exponential growth.
  2. Calculate percent rate of change: To find the percent rate of change, we need to subtract 11 from the base and then convert it to a percentage.\newlineThe percent rate of change is calculated as ((base−1)×100)%((\text{base} - 1) \times 100)\%.\newlineFor the given function, the base is (32)(\frac{3}{2}).
  3. Subtract 11 from base: Now we calculate the percent rate of change: ((32)−1)×100%((\frac{3}{2}) - 1) \times 100\%. First, subtract 11 from (32)(\frac{3}{2}): (32)−1=(32)−(22)=(12)(\frac{3}{2}) - 1 = (\frac{3}{2}) - (\frac{2}{2}) = (\frac{1}{2}).
  4. Convert to percentage: Next, convert (12)(\frac{1}{2}) to a percentage: (12)×100%=50%(\frac{1}{2}) \times 100\% = 50\%. The percent rate of change for the function is 50%50\%.

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