Restricting a domain to form inverse functions
Restrict a function's domain to make it one-to-one so its inverse exists: identify the portion passing the horizontal line test and define the inverse on that restricted interval.
Key Concepts
If a function's inverse is not a function itself, we can restrict the domain of the original function. This ensures that the new, smaller piece of the graph will pass the horizontal line test, making its inverse a proper function.
For $f(x) = x^2+5$, its full inverse isn't a function. If we restrict the domain to $x \ge 0$, the inverse becomes $f^{ 1}(x) = \sqrt{x 5}$, which is a function.: If we restrict the domain of $f(x) = (x 2)^2$ to $x \le 2$, its inverse becomes the function $f^{ 1}(x) = 2 \sqrt{x}$.
Some functions have inverses that are messy and not functions (like $y=x^2$). To fix this, we can perform surgery! By chopping off part of the original function's domain, we create a well behaved piece whose inverse is also a function. For a U shaped parabola, we can just keep one side of it, for example, everything where $x \ge 0$.
Common Questions
Why must you restrict a domain to form an inverse function?
An inverse function requires the original to be one-to-one, meaning each y-value corresponds to exactly one x-value. Functions like f(x)=x^2 fail the horizontal line test on their full domain. Restricting to x>=0 makes it one-to-one so an inverse can be defined.
How do you determine which domain restriction to use?
Choose an interval on which the function is monotonically increasing or decreasing. For f(x)=x^2, restricting to x>=0 keeps the increasing right branch. The restriction must preserve the useful range of the function for the application at hand.
What is the relationship between the restricted domain and the inverse's range?
The restricted domain of the original function becomes the range of the inverse function, and the range of the original on that restricted domain becomes the domain of the inverse. This reciprocal relationship is the defining property of inverse functions.