Find $$ f^{-1}(x) $$ for the following function: $$ f(x)=\frac{1}{x-4}-6 $$ $$ f^{-1}(x)=\square $$

Question

UpStudy AI · Accepted Answer

Let $$ y = f(x) = \frac{1}{x-4} - 6 $$.

1. Add 6 to both sides:
   $$
   y + 6 = \frac{1}{x-4}
   $$

2. Take the reciprocal of both sides (assuming $$ y \neq -6 $$):
   $$
   x - 4 = \frac{1}{y+6}
   $$

3. Solve for $$ x $$ by adding 4:
   $$
   x = \frac{1}{y+6} + 4
   $$

4. Replace $$ y $$ with $$ x $$ to express the inverse function:
   $$
   f^{-1}(x) = \frac{1}{x+6} + 4
   $$
$$ f^{-1}(x) = \frac{1}{x+6} + 4 $$

Find \( f^{-1}(x) \) for the following function: \( f(x)=\frac{1}{x-4}-6 \) \( f^{-1}(x)=\square \)