Graphing is a great tool that helps us understand how different functions work together. Here are some ways graphing makes this concept easier to grasp:

1. Understanding Function Behavior: When we graph single functions, we can see how they act. For example, if we have a line graph for $f(x)$ and a curve graph for $g(x)$, looking at both graphs helps us understand their relationship when we put them together.

2. Visualizing Composition: The way we combine functions is called composition, written as $(f \circ g)(x) = f(g(x))$. Graphing shows us how the result from one function is used as the starting point for the next. For instance, if $g(x) = x^2$ and $f(x) = 2x + 3$, the graph of $g(x)$ will show how we change $x$ into $x^2$. Then, the graph of $f(g(x))$ will show how those squared numbers are further changed.

3. Finding Points of Intersection: When we graph functions on their own and together, we can see where they cross. These crossing points can show solutions to equations like $f(g(x)) = k$, where $k$ is a fixed number.

4. Analyzing Changes: By graphing, we can see how changes affect the function compositions. For example, if $g(x)$ moves up or down, we can explore how that affects $f(g(x))$. This helps us understand how transformations work.

5. Estimating Outputs: Graphs let us visually estimate results from combined functions. For example, they help us see what $(f \circ g)(x)$ equals for certain input values, making it simpler to understand the overall behavior and connections.

In summary, graphing helps us understand how functions combine by giving us a visual way to connect the math concepts. This tool is especially helpful for Grade 12 students studying Algebra I.