Practice with Composed Functions: Building Graphs via Function Stacking

This example demonstrates how to form a new function by composing two functions and then visualize the resulting graph. It focuses on the key phenomenon behind function composition: applying an inner transformation to the input first (g), and then applying an outer function to that output (f), producing h(x)=f(g(x)). The main mathematical ideas include function composition, mapping effects on graphs, and interpreting how the combined rule changes the shape and location of the graph compared with the component functions. In GeoGebra, the construction is centered on defining the composite function h(x) from f and g, then plotting h and optionally plotting f and g for visual comparison. Because the provided GeoGebra “construction summary” contains no explicit commands or objects, the workflow is described in terms of the standard approach: define functions, create the composite function, and use graphing to compare. Suitable for secondary-level advanced students and introductory college learners studying function operations, transformation interpretation, and graph-based reasoning.