Digraphs of Functions

336. When we draw the digraph of a function f, we draw an arrow from

the vertex representing x to the vertex representing f(x). One of the relations you considered in Problems 333 and 334 is the relation of a function.

2_{We could imagine a digraph on an infinite set, but we could never draw all the vertices}

and edges, so people sometimes speak of digraphs on infinite sets. One just has to be more careful with the definition to make sure it makes sense for infinite sets.

A.1. RELATIONS AS SETS OF ORDERED PAIRS 151 (a) Which relation is the relation of a function?

(b) How does the digraph help you visualize that one relation is a function and the other is not?

337. Digraphs of functions help us to visualize whether or not they are onto or one-to-one. For example, let bothSandT be the set{−2,−1,0,1,2} and let S0 and T0 be the set{0,1,2}. Letf(x) = 2− |x|.

(a) Draw the digraph of the functionf assuming its domain isS and its range is T. Use the digraph to explain why or why not this function maps S ontoT.

(b) Use the digraph of the previous part to explain whether or not the function is one-to one.

(c) Draw the digraph of the functionf assuming its domain isS and its range is T0. Use the digraph to explain whether or not the function is onto.

(d) Use the digraph of the previous part to explain whether or not the function is one-to-one.

(e) Draw the digraph of the functionf assuming its domain isS0 and its range is T0. Use the digraph to explain whether the function is onto.

(f) Use the digraph of the previous part to explain whether the func- tion is one-to-one.

(g) Suppose that the function f has domain S0 and rangeT. Draw the digraph of f and use it to explain whetherf is onto.

(h) Use the digraph of the previous part to explain whetherf is one- to-one.

A one-to-one function from a set X onto a set Y is frequently called a bijection, especially in combinatorics. Your work in Problem 337 should show you that a digraph is the digraph of a bijection fromX toY

• if the vertices of the digraph represent the elements of X and Y, • if each vertex representing an element ofXhas one ond only one arrow

leaving it, and

• each vertex representing an element of Y has one and only one arrow entering it.

338. If we reverse all the arrows in the digraph of a bijection f, we get the digraph of another function g. Is g a bijection? What is f(g(x))? What is g(f(x))?

If f is a function from S to T, if g is a function from T to S, and if f(g(x)) =x for each x in T and g(f(x)) = x for each x inS, then we say thatg is an inverse off (andf is an inverse ofg).

More generally, if f is a function from a set R to a set S, and g is a function from S to T, then we define a new function f ◦g, called the

composition of f and g, by f ◦g(x) = f(g(x)). Composition of functions is a particularly important operation in subjects such as calculus, where we represent a function like h(x) = √x2_{+ 1 as the composition of the square} root function and the square and add one function in order to use the chain rule to take the derivative ofh.

The function ι (the Greek letter iota is pronounced eye-oh-ta) from a set S to itself, given by the rule ι(x) = x for every x in S, is called the

identity function on S. If f is a function from S to T and g is a function from T to S such that g(f(x)) = x for every x in S, we can express this by saying that g◦f =ι, where ιis the identity function of S. Saying that f(g(x)) = x is the same as saying that f ◦g = ι, where now ι stands for the identity function onT. We use the same letter for the identity function on two different sets when we can use context to tell us on which set the identity function is being defined.

339. If f is a function from S to T and g is a function from T to S such that g(f(x)) = x, how can we tell from context that g ◦f is the identity function on S and not the identity function on T? Online hint.

340. Explain why a function that has an inverse must be a bijection. 341. Is it true that the inverse of a bijection is a bijection?

342. If g andh are inverses of f, then what can we say aboutg and h? 343. Explain why a bijection must have an inverse.

Since a function with an inverse has exactly one inverse g, we callg the

inverse off. From now on, whenf has an inverse, we shall denote its inverse by f−1. Thus f(f−1(x)) =x and f−1(f(x)) =x. Equivalently f◦f−1 =ι andf−1◦f =ι.