(3 points) Determine whether each of these functions f : {a, b, c, d} → {a, b, c, d} is one-to-one and whether each of these functions is onto: (a) f(a) = b, f(b) = a, f(c) = c, f(d) = d (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c (c) f(a) = d, f(b) = b, f(c) = c, f(d) = d (2) (3 points) Determine whether each of these functions f : R → R is a one-to-one correspondence (i. e., onto and one-t

Step-by-step explanation:

Recall the following. Given a function f we say that  

f is one to one if for x,y in the domain of f and f(x)=f(y) then x=y.

This means that there are no different elements in the domain of the function  that are linked to the same value.

f is onto if for every y in the codomain of f, then there is an element x in the domain of f such that f(x) = y. That is, given an element y in the codomain of f, there exists and element in the domain that is linked to y.

a)  

f(a) =b, f(b)=a, f(c) = c, f(d) = d. Note that this function links each element in {a,b,c,d} to different elements in {a,b,c,d}, so it is a one to one function. Since for every element x in {a,b,c,d} there is an element y in {a,b,c,d} such that f(y) = x, then this function is also onto.  

b)  

f(a)=b, f(b) = b, f(c) = d, f(d) = c. Note that the elements a,b are linked both to b, so this function fails to be one-to-one. Also, note that the element a has no element x in {a,b,c,d} such that f(x) = a, so this function fails to be onto.  

c)  

f(a) = d, f(b) = b, f(c) = c, f(d) = d.  

As in b), since a and d are linked to d, then this function is not one to one. Also, since the element a has no element x in {a,b,c,d} such that f(x) = a, then it also fails to be onto.  

walter needs to make at least 5 more cookies but no more than 40. 5 ≥ x ≥ 40;

step-by-step explanation:

walter has to make a minimum of 20 cookies. he already has 15 cookies made, so he must make 5 more to meet the minimum.

the maximum amount of cookies he can make is 55. he already has 15, so he has to make no more than 40 more cookies to meet the maximum.

sorry i need the points

step-by-step explanation:

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