To find the inverse of the function \( f(x) = -0.085x + 2.96 \), we need to follow these steps:
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Replace \( f(x) \) with \( y \): \[ y = -0.085x + 2.96 \]
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Swap \( x \) and \( y \): \[ x = -0.085y + 2.96 \]
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Solve for \( y \):
- Start by isolating \( -0.085y \): \[ -0.085y = x - 2.96 \]
- Now divide by -0.085: \[ y = \frac{x - 2.96}{-0.085} \]
- This can be simplified: \[ y = -\frac{x - 2.96}{0.085} = -\frac{x}{0.085} + \frac{2.96}{0.085} \]
- Now calculate \( \frac{2.96}{0.085} \): \[ \frac{2.96}{0.085} \approx 34.8235 \]
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Putting it all together: \[ f^{-1}(x) = -\frac{x}{0.085} + 34.8235 \]
Thus, the inverse function is: \[ f^{-1}(x) = -\frac{1}{0.085}x + 34.8235 \] To express this as a decimal and simplify: \[ f^{-1}(x) \approx -11.7647x + 34.8235 \]
Now, let's summarize the output representation:
- The outputs of the inverse function \( f^{-1}(x) \) represent the number of years after 2000 when the percentage of children taking antidepressants is \( x % \).
Part b: Find when the percentage is 2.4% using the inverse function:
We will set \( x = 2.4 \) and use the inverse function: \[ f^{-1}(2.4) = -\frac{1}{0.085}(2.4) + 34.8235 \] Calculating this:
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First compute: \[ -\frac{1}{0.085} \cdot 2.4 \approx -28.2353 \]
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Then add this to 34.8235: \[ f^{-1}(2.4) \approx -28.2353 + 34.8235 \approx 6.5882 \]
Thus, \( f^{-1}(2.4) \approx 6.59 \).
Interpretation:
- This means that the percentage of children taking antidepressants reached 2.4% approximately in the year 2000 + 6.59, which is around the year 2007 (to the nearest year).
So, you would write:
- a. The inverse function is \( f^{-1}(x) = -\frac{1}{0.085}x + 34.8235 \).
- b. The percentage of children taking antidepressants was 2.4% approximately in the year 2007.
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