Questions on compositions of functions
In this worksheet you will be tasked with finding examples, counterexamples and analysing whether some statements about the compostion of functions are true or not. You can use the applet below to enter some functions and check their graphs in order to help you check your answers.
Answers are not provided, as there are many possible solutions, instead a hint is given to help you come up with your own answer.
Find two functions such that their composition is 1-1.
Find two functions such that their composition is non-decreasing.
Find two functions such that their composition is onto.
Find two functions such that their composition is not a polynomial.
Find two functions such that their composition is both 1-1 and onto.
Find two functions f(x), g(x) such that f(g(x)) is 1-1 but g(f(x)) is not.
Give a counterexample to the claim that no composition of functions can be 1-1.
Give a counterexample to the claim that for two functions f(x),g(x) we always have f(g(x)) = g(f(x)).
Give an example of two functions f(x),g(x), where the graph y=f(g(x)) intersects the graph y=g(f(x)).