Find $$ f^{-1}(x) $$ for the following function: $$ f(x)=\frac{4}{x+2}+1 $$ $$ f^{-1}(x)= $$

Question

UpStudy AI · Accepted Answer

1. Start with the function:
   $$
   f(x) = \frac{4}{x+2} + 1
   $$
   Replace $$ f(x) $$ with $$ y $$:
   $$
   y = \frac{4}{x+2} + 1
   $$

2. Isolate the fraction by subtracting 1 from both sides:
   $$
   y - 1 = \frac{4}{x+2}
   $$

3. Multiply both sides by $$ x+2 $$ to eliminate the fraction:
   $$
   (y - 1)(x + 2) = 4
   $$

4. Solve for $$ x + 2 $$ by dividing both sides by $$ y - 1 $$ (note that $$ y \neq 1 $$):
   $$
   x + 2 = \frac{4}{y - 1}
   $$

5. Isolate $$ x $$ by subtracting 2 from both sides:
   $$
   x = \frac{4}{y - 1} - 2
   $$

6. Swap $$ x $$ and $$ y $$ to find the inverse function:
   $$
   y = \frac{4}{x - 1} - 2
   $$

Thus, the inverse function is:
$$
f^{-1}(x) = \frac{4}{x-1} - 2
$$
$$ f^{-1}(x) = \frac{4}{x-1} - 2 $$

Find \( f^{-1}(x) \) for the following function: \( f(x)=\frac{4}{x+2}+1 \) \( f^{-1}(x)= \)