Sum, composition and inverse of functions
Add two functions, keep the order in a composite function, find the inverse of a machine by undoing its steps, build the inverse graph by reflecting in y = x, and use a horizontal line to see which functions have an inverse.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- UK: Year 10
- Germany: grade 10
- Iran: grade 11
- Afghanistan: grade 10
- Syria: grade 11
- Palestine: grade 9
- Ukraine: grade 11
- Russia: grade 10
Guide
(f + g)(x) means f(x) + g(x): add the values of the two functions at the same x. Products and quotients work the same way; for a quotient g(x) must not be zero.
In (g∘f)(x) = g(f(x)) f is done first and then g. The order matters: the machine "× 2, then + 1" is not "+ 1, then × 2".
To find an inverse function undo the steps in reverse order. The graph of f−1 is the reflection of the graph of f in the line y = x: the point (a, b) goes to (b, a). Only a function that gives each output once (one-to-one function, the horizontal-line test) has an inverse.