To determine if a function or a matrix has an inverse, we examine the following criteria:

1. Inverse of a Function: A function has an inverse if it is bijective (both one-to-one and onto). This requires:

   • One-to-One (Injective): Each input maps to a unique output.
   • Onto (Surjective): Every possible output is covered by the function.

   A practical test for functions of a single variable is the Horizontal Line Test: If every horizontal line intersects the function’s graph at most once, the function is one-to-one, and hence, it has an inverse.

2. Inverse of a Matrix: A square matrix (with the same number of rows and columns) has an inverse if it is nonsingular or invertible, meaning:

   • The matrix’s determinant is not zero.
   • The matrix has full rank (the rank is equal to the matrix’s dimension).

Would you like further details on determining inverses or on any specific type of function or matrix? Here are some related questions for deeper exploration:

1. What is the Horizontal Line Test, and how does it work?
2. How does one find the inverse of a given function?
3. How can we determine if a matrix is invertible?
4. What is the relationship between the determinant and the invertibility of a matrix?
5. How do inverses apply to complex functions or matrices?

Tip: To check if a matrix is invertible quickly, calculate its determinant—if it’s zero, the matrix has no inverse.