Graph the function $f\left(x\right)=\{\begin{array}{c}x\text{ if }x\le 0\\ x+1\text{ if }x>0\end{array}$.

Verified step by step guidance

1. Step 1: Understand the piecewise function. The function f(x) is defined in two parts: f(x) = x for x ≤ 0 and f(x) = x + 1 for x > 0.
2. Step 2: Graph the first part of the function, f(x) = x, for x ≤ 0. This is a straight line through the origin with a slope of 1, but only for x-values less than or equal to 0.
3. Step 3: Graph the second part of the function, f(x) = x + 1, for x > 0. This is a straight line with a slope of 1, starting at the point (0, 1) and continuing for x-values greater than 0.
4. Step 4: Identify the point of transition at x = 0. For x = 0, the function value is 0 from the first part, so the point (0, 0) is included in the graph.
5. Step 5: Combine the two parts on the same set of axes. Ensure the graph is continuous at x = 0, with a closed circle at (0, 0) and an open circle at (0, 1) to indicate the transition between the two parts.