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Inverse Functions

Domain and Range
  • Domain is the input into a function
  • Range is the output of the function
Domain and range in inverse functions
  • Domain of \(f()\) is the range of \(f^{-1}()\)
  • Range of \(f()\) is the domain of \(f^{-1}()\)
\(3x-5\), the flow method.
  • \(x\underbrace{ \to }_{ \times3 }3x\underbrace{ \to }_{ -5 }3x-5\)
  • \(\frac{y-5}{3}\underbrace{ \to }_{ \times3 }y-5\underbrace{ \to }_{ +5 }y\)
\(\frac{3}{x-1}\), changing the variable
  • Start with \(y=\frac{3}{x-1}\) and trying to find the inverse.
  • \(x=\frac{3}{y-1}\)

  • Rearrange for \(y\):
  • \(y-1=\frac{3}{x}\)
  • \(y=\frac{3}{x}+\frac{1}{1}\)
  • \(y=\frac{3+x}{x}\)
  • \(f^{-1}(x)=\frac{3+x}{x}\)
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