Step 1 :Replace \(h(x)\) with \(y\) to get \(y = \frac{4x - 5}{x + 9}\)
Step 2 :Swap \(x\) and \(y\) to find the inverse: \(x = \frac{4y - 5}{y + 9}\)
Step 3 :Solve for \(y\) to get \(xy + 9x = 4y - 5\)
Step 4 :Rearrange the equation to get \(xy - 4y = -9x - 5\)
Step 5 :Factor out \(y\) to get \(y(x - 4) = -9x - 5\)
Step 6 :Divide by \(x - 4\) to get \(y = \frac{-9x - 5}{x - 4}\)
Step 7 :So, the inverse function is \(h^{-1}(x) = \frac{-9x - 5}{x - 4}\)
Step 8 :The domain of \(h^{-1}(x)\) is all real numbers except 4, so the domain is \((-\infty, 4) \cup (4, \infty)\)
Step 9 :The range of \(h^{-1}(x)\) is all real numbers except -9, so the range is \((-\infty, -9) \cup (-9, \infty)\)
Step 10 :\(\boxed{h^{-1}(x) = \frac{-9x - 5}{x - 4}}\)
Step 11 :Domain of \(h^{-1}\) is \(\boxed{(-\infty, 4) \cup (4, \infty)}\)
Step 12 :Range of \(h^{-1}\) is \(\boxed{(-\infty, -9) \cup (-9, \infty)}\)