Determining Restricted Domains for Function Inverses: y x^2 3
To find a restricted domain for the function y x^2 3 such that its inverse is also a function, we first need to understand the concept of a function being one-to-one (injective).
Understanding Function Inverses and One-to-One Functions
A function must pass the one-to-one (injective) criterion to have an inverse that is also a function. A function is one-to-one if each input corresponds to a unique output, and no two different inputs give the same output. This property is often tested using the horizontal line test: a function is one-to-one if any horizontal line intersects the graph of the function at most once.
Analysis of y x^2 3
The function y x^2 3 is a parabola that opens upwards. Due to its symmetric nature, it fails the one-to-one test. For example, the value of the function is the same for both positive and negative x values. Specifically, f(-2) f(2) 7, indicating that the function is not one-to-one over its entire domain.
Restricting the Domain to Ensure Injectivity
To ensure that the inverse of the function is also a function, we need to restrict the domain to make the function one-to-one. This can be accomplished by either considering the right half or the left half of the parabola.
Right Half of the Parabola
- Domain: [0, infty)
- Inverse Function: The inverse function would then be y sqrt{x - 3}.
Left Half of the Parabola
- Domain: (-infty, 0]
- Inverse Function: The inverse function would be y -sqrt{x - 3}.
Conclusion
In summary, the restricted domains that make the inverse a function are:
Domain: [0, infty) Domain: (-infty, 0]Choosing either of these restrictions ensures that the function is one-to-one, and consequently, that its inverse is also a function. It is important to note that there are infinitely many ways to restrict the domain to achieve this. By selecting a domain that makes the function one-to-one, we also ensure that the range of the original function and the domain of the inverse function are correctly defined.
Additional Considerations
There are technically many different ways to restrict the domain to make the inverse a function. However, the most common and practical choice is to restrict the domain to the left or right side of the vertex of the parabola. This restriction ensures the one-to-one property and allows for a well-defined inverse function.
Remember, any domain restriction must be carefully chosen to avoid any horizontal line intersecting the graph of the function at more than one point, thereby failing the one-to-one test.