Question

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Part 1
The function that models the percent of children taking antidepressants from 2004 to 2009 is ​f(x)equalsnegative 0.085 x plus 2.96​, where x is the number of years after 2000.
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 2.4​%.
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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 \( f(x) = -0.085x + 2.96 \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = -0.085x + 2.96
\]

2. **Swap \( x \) and \( y \)**:
\[
x = -0.085y + 2.96
\]

3. **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 \( \frac{2.96}{0.085} \):
\[
\frac{2.96}{0.085} \approx 34.8235
\]

4. **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:
1. First compute:
\[
-\frac{1}{0.085} \cdot 2.4 \approx -28.2353
\]

2. 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.