Forgetting to reflect the correct portion of the graph about the x-axis when dealing with |f(x)| or misunderstanding the transformations involved with f(|x|).
Break down the transformation step-by-step, starting with the innermost function g(x), and carefully consider how its range affects the domain of the outer function f(x).
Sketch the boundary curve first, then test points in different regions to determine which satisfy the inequality, shading the appropriate region.
Accurately representing the amplitude, period, phase shift, and vertical translation, paying close attention to the scale on both axes.
Misinterpreting the axes or units, and failing to relate the graphical features (e.g., maximum, minimum, intercepts) to the real-world quantities they represent.
Double-check the window settings, function input, and ensure you understand the calculators limitations, especially when dealing with asymptotes or discontinuities.