To find the inverse of a function, follow these steps:
- Replace the function notation f(x)f(x)f(x) with yyy.
- Swap the variables xxx and yyy in the equation.
- Solve the new equation for yyy to isolate yyy on one side.
- Replace yyy with the inverse notation f−1(x)f^{-1}(x)f−1(x) to denote the inverse function.
The inverse function essentially "undoes" the original function, such that applying the function and then its inverse returns the original input xxx. For example, to find the inverse of f(x)=2x−7f(x)=2x-7f(x)=2x−7:
- Start with y=2x−7y=2x-7y=2x−7.
- Swap variables: x=2y−7x=2y-7x=2y−7.
- Solve for yyy: y=x+72y=\frac{x+7}{2}y=2x+7.
- So the inverse function is f−1(x)=x+72f^{-1}(x)=\frac{x+7}{2}f−1(x)=2x+7.
Finally, verify your result by checking that f(f−1(x))=xf(f^{-1}(x))=xf(f−1(x))=x and f−1(f(x))=xf^{-1}(f(x))=xf−1(f(x))=x. This method applies to many types of functions, but beware that not all functions have inverses that are functions (e.g., some may fail the vertical line test after swapping variables). This summary combines several detailed explanations and examples from reliable educational sources.
