An inverse function is a function that reverses the action of another function. If a function f takes an input x and produces output y, then its inverse function f inverse takes y and produces the original input x. For example, if f doubles a number, then f inverse halves it.
To find an inverse function, we follow four steps. First, replace f of x with y. Second, swap x and y variables. Third, solve for y. Finally, replace y with f inverse of x. Let's see this with an example where f of x equals two x plus one.
For a function to have an inverse, it must be one-to-one. This means each output corresponds to exactly one input. We can test this using the horizontal line test. If any horizontal line intersects the graph more than once, the function is not one-to-one. For example, y equals x cubed is one-to-one, but y equals x squared is not.
The key property of inverse functions is that f of f inverse of x equals x, and f inverse of f of x equals x. This means applying a function and its inverse returns the original value. We can see this graphically where the function and its inverse are reflections across the line y equals x.
To summarize what we have learned about inverse functions: they reverse the action of original functions, functions must be one-to-one to have inverses, we use the horizontal line test to check this property, and the composition of a function and its inverse returns the original value. Inverse functions are fundamental concepts widely used in mathematics and science.