Functions: Compositions, Inverses, and Transformations
Lessons in this chapter
About Functions: Compositions, Inverses, and Transformations
Functions: Compositions, Inverses, and Transformations opens AP Precalculus by revisiting how functions combine, reverse, and move.
Composite Functions covers building a new function by feeding one function’s output into another as input.
Inverse Functions covers reversing a function’s input-output pairing exactly, undoing what the original function does.
Transformations closes the chapter with the shifts, stretches, and reflections that move a function’s graph without changing its underlying shape.
The three lessons, Composite Functions, Inverse Functions, and Transformations, each open as a practice worksheet worked in the page. The problems are scored as you answer them, so a composition built in the wrong order is caught at that question rather than two steps later. Free to try, with no login needed for your first 3.
Frequently asked questions
Composite Functions builds one function out of two, f(g(x)), and works out what the domain of the result can be. Inverse Functions runs a function backwards, finds the inverse algebraically, and checks it both by composition and by reflection in the line y = x. Transformations covers the shifts, stretches, and reflections that move a graph without changing the family it belongs to. This sits in the Functions (F) conceptual category of the high school Common Core, and it is the toolkit the rest of the AP Precalculus course framework assumes you already have.
Function composition, written (f ∘ g)(x) or f(g(x)), means evaluate g first, then use that result as the input to f.
An inverse function reverses a function's input-output pairing exactly: if f(a) = b, then the inverse of f maps b back to a.
Changing a function’s equation shifts, stretches, compresses, or reflects its graph, without changing the function’s basic underlying shape.
Only a one-to-one function (one that passes the horizontal line test) has a true inverse function: otherwise the reversed pairing isn't a function at all.
Confusing (f ∘ g)(x) with (g ∘ f)(x): composition is generally not commutative, so the order the two functions are combined in changes the result.