How to tell if a function and a composite function is onto or one to one

Question

For each of the following, f : A → B, g : B → C. Which one are true and which ones are false? So far i have,

1. f is onto but g ◦ f is not onto. (False)
2. f is 1-1 but g ◦ f is not 1-1. (False)
3. g is onto but g ◦ f is not onto.(True)
4. g is 1-1 but g ◦ f is not 1-1. (True)
5. f is not onto but g ◦ f is onto. (true)
6. f is not 1-1 but g ◦ f is 1-1. (false)
7. g is not 1-1 but g ◦ f is 1-1. (true)

I just wanted to double check if I had the concept of one to one or onto functions right, and if i have made any mistakes could you please give an example why ?

Thanks

Answer 1

I found it useful during learning stages to think of $f\colon A\to B$ in the following way. $A$ and $B$ are some states consisting of some towns.A function $f$ is simply a plan that specifies roads starting from each city of $A$ and ending in some city of $B$. (another function $g$ is a different scheme of road plans).In this analogy it is possible that roads starting from two different towns of $A$ may end in the same town of $B$. But if this never happens then $f$ is one-to-one.

And if every town of the state $B$ can be reached from some town in $A$ by the plan specified in $f$ then the function is onto. Now convert all your questions into these scheme of road (bringing in a third state $C$). Always remember that a road plan must have roads starting from EVERY town in its starting state and has only one terminating town.