To graph the composite function h(x) = f(g(x)) using the graphs of f and g, you’ll follow a step-by-step approach that ensures you’re capturing the necessary transformations accurately. Here’s how to do it:
Step 1: Understand your functions
Make sure you have the graphs of both functions f(x) and g(x). Identify the segments of these graphs you want to focus on before proceeding to graph h. If your interest lies within certain closed intervals, take note of the corresponding x-values.
Step 2: Determine the range of g(x)
Calculate the output of g(x) over the interval you’re interested in. This helps to establish the vertical distance that g travels as x varies. For example, if you’re looking at [a, b], evaluate g(a) and g(b) to find the minimum and maximum values of the range.
Step 3: Evaluate f at g(x)
Once you have the range of g, the next step is to substitute these values into f. This means you will calculate h(x) as h(x) = f(g(x)) for each x in your original interval. Keep this in mind for the closed endpoints: evaluate h(a) and h(b) to define the endpoints of your graph.
Step 4: Draw h(x)
Finally, plot the points (a, h(a)) and (b, h(b)) on a coordinate system. You’ll also want to sketch the curve by connecting the dots, being mindful of the behavior of f as it relates to the outputs from g. Since you’re interested in closed segments, ensure to include solid dots at (a, h(a)) and (b, h(b)) to represent that these endpoints are included in your final graph.
Example
For example, if you have f(x) = x^2 and g(x) = 2x – 1, and you’re interested in the interval [1, 3]:
- Calculate g(1) = 1 and g(3) = 5.
- Then, substitute these values into f: h(1) = f(g(1)) = f(1) = 1^2 = 1 and h(3) = f(g(3)) = f(5) = 5^2 = 25.
Your points are (1, 1) and (3, 25). Draw a segment connecting these, making sure to mark it closed at both ends.
Conclusion
By following these steps, you can effectively utilize the graphs of f and g to plot the composite function h(x) with closed segments, making your graph both accurate and informative.