Question

Determine whether each of these functions from \(\left\{ {a,b,c,d} \right\}\) to itself is one-to-one.

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

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

Short answer

1. One-to-one
2. Not one-to-one
3. Not one-to-one

Step-by-step solution

Step: 1

a) By definition of the one-to-one function for each y value, there should be exactly one corresponding value of x.

Here we have obtained a relationship between the dependent and independent variables which satisfy the definition of a one-to-one function.

Hence the given function is a one-to-one function.

Step: 2

b) By definition of the one-to-one function for each y value, there should be exactly one corresponding value of x.

Here we have obtained a relationship between the dependent and independent variables which does not satisfy the definition of a one-to-one function.

Hence the given function is not a one-to-one function.

Step: 3

c) By definition of the one-to-one function for each y value, there should be exactly one corresponding value of x.

Here we have obtained a relationship between the dependent and independent variables which does not satisfy the definition of a one-to-one function.

Hence the given function is not a one-to-one function.