The function that models the percent of children taking antidepressants from 2004 to 2009 is ​f(x)equalsnegative 0.082 x plus 2.98​, where x is the number of years after 2000.

Question

a. Find the inverse of this function. What do the outputs of the inverse function​ represent?
b. Use the inverse function to find when the percentage is 2.6​%.
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Part 1
a. The inverse function is f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
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Answer 1

To find the inverse of the function \( f(x) = -0.082x + 2.98 \), we will follow the steps to solve for \( x \) in terms of \( y \).

1. Start with the original equation: \[ y = -0.082x + 2.98 \]

2. Swap \( x \) and \( y \) to find the inverse: \[ x = -0.082y + 2.98 \]

3. Solve for \( y \): \[ x - 2.98 = -0.082y \]

   \[ y = \frac{2.98 - x}{0.082} \]

Thus, the inverse function is: \[ f^{-1}(x) = \frac{2.98 - x}{0.082} \]

Outputs of the inverse function: The outputs of the inverse function represent the number of years after 2000 when the percentage of children taking antidepressants is \( x % \).

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For part b, we need to find when the percentage is \( 2.6% \). We will substitute \( x = 2.6 \) into the inverse function:

\[ f^{-1}(2.6) = \frac{2.98 - 2.6}{0.082} \]

Calculating this gives:

\[ f^{-1}(2.6) = \frac{0.38}{0.082} \approx 4.6341 \]

Since \( x \) represents the number of years after 2000, we can interpret this as approximately \( 4.6341 \) years after the year 2000.

Thus, the approximate year is:

\[ 2000 + 4.6341 \approx 2004.63 \]

This means that the percentage of children taking antidepressants was \( 2.6% \) around mid-2004.