Invertible functions

When studying "Determining if a function is invertible," focus on these key points:

1. Definition of Invertibility: A function is invertible if there exists another function that can reverse its effect.

2. One-to-One (Injective) Function: A function must be one-to-one to be invertible. This means that each output must correspond to exactly one input.

3. Horizontal Line Test: A graphical method to determine injectivity—if any horizontal line intersects the graph of the function more than once, it is not one-to-one.

4. Domain and Range: Ensure that the domain of the original function matches the range of the inverse function, and vice versa.

5. Algebraic Methods: If applicable, solve for the input variable explicitly to see if a unique output exists for each input.

6. Bijective Functions: A function is invertible if it is both injective (one-to-one) and surjective (onto).

7. Formal Inverse Notation: The notation $f^{-1}(y)$ represents the inverse function, which yields the original input for a given output.

8. Applications and Practicality: Understanding the context in which a function is needed to be invertible, such as in solving equations or modeling scenarios.

By mastering these concepts, you can effectively determine the invertibility of a function.

When studying "Restricting domains of functions to make them invertible," key points include:

1. Understanding Invertibility: A function is invertible if each output corresponds to exactly one input. This necessitates a one-to-one (bijective) relationship.

2. Definition of Domain Restriction: To make a function invertible, you can limit its domain (the set of inputs) to a subset where it maintains a one-to-one relationship.

3. Identifying One-to-One Intervals: Analyze the function's behavior (e.g. monotonicity) to determine intervals where it is either strictly increasing or strictly decreasing, which helps in finding suitable restrictions.

4. Graphical Representation: Use graphs to visualize functions and their potential inverse functions. A horizontal line test can indicate where the original function is one-to-one.

5. Creating Inverses: Once a function is restricted, finding its inverse involves swapping the dependent and independent variables and solving for the new dependent variable.

6. Notation: Understand the notation for functions and their inverses, commonly $f^{-1}(x)$ for the inverse of function $f(x)$.

7. Practical Examples: Work through examples of quadratic, cubic, and trigonometric functions to see how restriction alters their domains to allow for invertibility.

8. Applications: Recognize scenarios where domain restriction is useful, such as in calculus and solving equations where inverse operations are needed.

9. Pitfalls: Be cautious about overly restricting the domain, which may lead to omitting essential solutions or introducing discontinuities in the function.

By mastering these points, you will gain a clear understanding of how to restrict domains effectively to create invertible functions.