To graph the composite function h(x) = f(g(x)), it’s essential to first understand the individual functions f and g and how they interact.
1. **Identify the Graphs**: Start by examining the graphs of f(x) and g(x). Knowing the behavior of these functions will help you visualize how they will combine.
2. **Determine g(x) Values**: For each x-value on the graph, find the corresponding g(x) value. This means you will trace along the x-axis to see where the point intersects the graph of g.
3. **Use g(x) Values in f(x)**: Once you have g(x), apply these values to the function f. So for every point where you have an x, you will calculate f(g(x)).
4. **Plot h(x)**: Use the calculated values from f(g(x)) to plot the new function h. For each x-value, you will now find a new point on the graph that corresponds to h(x).
5. **Graph Segments**: When graphing, ensure that you connect the points with line segments, remembering that both f and g are defined only on their respective intervals. This may create segments in the graph of h that can only be drawn within the domain defined by the outputs of g applicable to the inputs of f.
6. **Consider Closed Endpoints**: If the problem specifies closed endpoints, make sure to include these endpoints on your graph to denote where the functions start and stop. This is crucial as it indicates that those points are included in the function’s range.
By following these steps, you can accurately graph h(x) = f(g(x)). Remember to take your time, verify your points, and examine the nature of your functions to ensure a smooth composite graph.