Park Weber
09/08/2024 · High School
Let \( f(x)=\frac{1}{x+8} \) \( f^{-1}(x)= \)
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Tutor-Verified Answer
Step-by-step Solution
To find the inverse of the function \( f(x) = \frac{1}{x+8} \), we will follow these steps:
1. Replace \( f(x) \) with \( y \):
\[
y = \frac{1}{x + 8}
\]
2. Swap \( x \) and \( y \):
\[
x = \frac{1}{y + 8}
\]
3. Solve for \( y \):
- First, multiply both sides by \( y + 8 \):
\[
x(y + 8) = 1
\]
- Expand the left side:
\[
xy + 8x = 1
\]
- Rearrange to isolate \( y \):
\[
xy = 1 - 8x
\]
- Divide both sides by \( x \):
\[
y = \frac{1 - 8x}{x}
\]
4. Thus, the inverse function is:
\[
f^{-1}(x) = \frac{1 - 8x}{x}
\]
To simplify further, we can rewrite it as:
\[
f^{-1}(x) = \frac{1}{x} - 8
\]
So, the final answer is:
\[
\boxed{\frac{1 - 8x}{x}} \quad \text{or} \quad \boxed{\frac{1}{x} - 8}
\]
Quick Answer
\( f^{-1}(x) = \frac{1 - 8x}{x} \) or \( f^{-1}(x) = \frac{1}{x} - 8 \)
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