To determine which function corresponds to the given graph, let's analyze each piecewise function option in relation to the graph's behavior.

Analysis of the Graph

The graph shows:

1. For $x \leq 1$, the function has an increasing curve, likely quadratic or linear, that opens upwards.
2. At $x = 1$, there is a defined point.
3. For $x > 1$, there is a downward sloping line