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