1. The problem asks: When does a function have an inverse function? 2. A function $f$ has an inverse function $f^{-1}$ if and only if $f$ is **one-to-one** (injective). 3. Being one-to-one means that for every $x_1$ and $x_2$ in the domain, if $f(x_1) = f(x_2)$ then $x_1 = x_2$. 4. This ensures that the function passes the **Horizontal Line Test**: no horizontal line intersects the graph of $f$ more than once. 5. If $f$ is one-to-one, then the inverse function $f^{-1}$ exists and satisfies: $$f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(y)) = y$$ 6. In summary, the key condition for a function to have an inverse is that it must be one-to-one (injective).