Functions, inverses and composition

Learning objectives

Use the language of functions correctly, identifying the domain and range of a given function and explaining in words why a mapping is or is not a function

Decide whether a function is one-one, using a horizontal line test on its graph, and explain why a function that is not one-one has no inverse over its full domain

Find the inverse of a one-one function by writing y = f(x), interchanging the variables and making the new y the subject, and state the domain and range of f⁻¹

Form composite functions such as fg(x) and f²(x), evaluate them at given values, and note that fg(x) and gf(x) are generally different

Sketch the graphs of y = f(x) and y = f⁻¹(x) on the same axes and explain that each is the reflection of the other in the line y = x

Relate the graph of y = |f(x)| to that of y = f(x) by reflecting in the x-axis every part of the curve that lies below it