Restricting the domain

Definition

Restricting the domain involves limiting the set of input values (x-values) for which a function is defined to ensure it meets certain criteria, such as being one-to-one. This is often necessary when finding the inverse of a function.

5 Must Know Facts For Your Next Test

1. Restricting the domain can make a non-injective function injective, allowing for an inverse to exist.
2. The restricted domain must include all x-values that allow the function to pass the horizontal line test.
3. Common functions that require domain restrictions for their inverses include quadratic and trigonometric functions.
4. When restricting the domain, it's important to specify both end points and open or closed intervals appropriately.
5. Graphically, restricting the domain often involves highlighting a specific portion of the original function's graph.

Review Questions

• Why is it necessary to restrict the domain of some functions in order to find their inverses?
• How does restricting the domain affect whether a function passes the horizontal line test?
• Provide an example of a function where restricting its domain allows it to have an inverse.

Related terms

Inverse Function: A function that reverses another function, such that if $f(x) = y$, then $f^{-1}(y) = x$.

Horizontal Line Test: A method used to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, then the function does not have an inverse.

One-to-One Function: A function where each output value is paired with exactly one input value, ensuring that every element in the range corresponds with one unique element in the domain.