Domain Restriction for Inverse Functions

When a function fails the horizontal line test, its domain can be restricted to create a one-to-one function that has an inverse function. The restricted domain should eliminate duplicate y-values while preserving the essential behavior of the original function..

Example: For f(x) = x^2, restrict the domain to x \geq 0 to create f(x) = x^2, x \geq 0, which has inverse f^{-1}(x) = \sqrt{x}

For f(x) = (x - 2)^2 + 1, restrict the domain to x \geq 2 to create a one-to-one function with inverse f^{-1}(x) = 2 + \sqrt{x - 1}

Domain restriction is a technique used when the original function is not one-to-one, meaning it fails the horizontal line test. By carefully choosing which portion of the domain to keep, we can eliminate the duplicate y-values that prevent the inverse from being a function.

Domain Restriction for Inverse Functions is a Grade 11 math topic covered in enVision, Algebra 2 in Chapter 5: Rational Exponents and Radical Functions. Students at this level study the concept as part of their grade-level standards and are expected to explain, analyze, and apply what they have learned.

By carefully choosing which portion of the domain to keep, we can eliminate the duplicate y-values that prevent the inverse from being a function. The key is to select a continuous interval where the function is strictly increasing or strictly decreasing. This restriction ensures that each output value corresponds to exactly one input value, making the inverse relation a true function..