# Functions

Solutions are due before 11:59 PM on **October 20, 2020 **.

- (2pts)

Determine whether the mappings from R to R shown below are or are not functions, and explain your decision.

- (2pts)

Determine whether each of the following functions from R to R is a bijection, and explain your decision. HINT: Plotting these functions may help you with your decision.

*f*(*x*) = −*x*^{2 }+ 2*f*(*x*) =*x*^{3 }−*x*^{2}

- (2pts)

Suppose the function *f *: *A *→ *B *is a bijection. What can you say about the values |*A*| and |*B*|?

- (2pts)

Let *f *: {1*,*2*,*3*,*4*,*5*,*6}→{*red,yellow,beige,green,umber,teal*} be a one-to-one function. Prove, by contradiction, that *f *is a bijection.

- (2pts) Let
*A*= {1*,*2*,*3*,*4}

Let *B *= {*a,b*} Let *C *= {*curling,hockey,table*–*tennis*}

- How many
*one-to-one*functions are there from*C*to*A*? - How many
*onto*functions are there from*C*to*B*?

(Hint: count the non-onto functions)

- (4pts)

Decide for each of the following expressions: Is it a function? If so,

(i) what is its domain, codomain, and image? (ii) is it injective? (why or why not)

(iii) is it surjective? (why or why not) (iv) is it invertible? (why or why not)

*f*: R→R defined by*x*7→*x*^{3}

1

*f*: R×Z→Z defined by (*r,z*) →d*r*e∗*z*

- (2pts)

Let *f *be a function from the set *A *to the set *B*. Let *S *and *T *be subsets of *A*. Show that

*f*(*S*∪*T*) =*f*(*S*) ∪*f*(*T*): (b)*f*(*S*∩*T*) ⊆*f*(*S*) ∩*f*(*T*).

- (1pts) find the inverse function of
*f*(*x*) =*x*^{3 }+ 1 - (2pts)

Suppose that *g *is a function from *A *to *B *and f is a function from *B *to *C*.

- Show that if both f and g are one-to-one functions, then
*f*◦*g*is also one-to-one. - Show that if both
*f*and*g*are onto functions, then*f*◦*g*is also onto.

- (1pts)

Find *f *◦*g *and *g *◦*f*, where *f*(*x*) = *x*^{2 }+1 and *g*(*x*) = *x*+2, are functions from R to R.

2