Find the inverse of $$ f(x)=\frac{x-3}{-2 x-5} $$ $$ f^{-1}(x)=\square $$

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Find the inverse of the function $$ f(x)=\frac{x-3}{-2x-5} $$
Function by following steps:
- step0: Find the inverse:
  $$f\left(x\right)=\frac{x-3}{-2x-5}$$
- step1: Simplify:
  $$f\left(x\right)=-\frac{x-3}{2x+5}$$
- step2: Replace $$f\left(x\right)$$ with $$y:$$
  $$y=-\frac{x-3}{2x+5}$$
- step3: Interchange the variables:
  $$x=-\frac{y-3}{2y+5}$$
- step4: Swap the sides:
  $$-\frac{y-3}{2y+5}=x$$
- step5: Rewrite the expression:
  $$\frac{-y+3}{2y+5}=x$$
- step6: Cross multiply:
  $$-y+3=\left(2y+5\right)x$$
- step7: Simplify the equation:
  $$-y+3=x\left(2y+5\right)$$
- step8: Expand the expression:
  $$-y+3=2xy+5x$$
- step9: Move the variable to the left side:
  $$-y+3-2xy=5x$$
- step10: Collect like terms:
  $$\left(-1-2x\right)y+3=5x$$
- step11: Move the constant to the right side:
  $$\left(-1-2x\right)y=5x-3$$
- step12: Divide both sides:
  $$\frac{\left(-1-2x\right)y}{-1-2x}=\frac{5x-3}{-1-2x}$$
- step13: Divide the numbers:
  $$y=\frac{-5x+3}{1+2x}$$
- step14: Replace $$y$$ with $$f^{-1}\left(x\right):$$
  $$f^{-1}\left(x\right) = \frac{-5x+3}{1+2x}$$
The inverse of the function $$ f(x)=\frac{x-3}{-2x-5} $$ is $$ f^{-1}(x) = \frac{-5x+3}{1+2x} $$.
$$ f^{-1}(x) = \frac{-5x+3}{1+2x} $$

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