Chapter 1.2, Problem 4E

For each of the following mappings $f:Z\to Z$, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers.

a. $f\left(x\right)=2x$ b. $f\left(x\right)=3x$

c. $f\left(x\right)=x+3$ d. $f\left(x\right)=x^3$

e. $f\left(x\right)=\left|x\right|$ f. $f\left(x\right)=x-\left|x\right|$

g. $f\left(x\right)=\{\begin{array}{cc}x& \text{if }x\text{is even}\\ 2x-1& \text{if }x\text{is odd}\end{array}$ h. $f\left(x\right)=\{\begin{array}{cc}x& \text{if }x\text{is even}\\ x-1& \text{if }x\text{is odd}\end{array}$

i. $f\left(x\right)=\{\begin{array}{cc}x& \text{if }x\text{is even}\\ \frac{x-1}{2}& \text{if }x\text{is odd}\end{array}$ j. $f\left(x\right)=\{\begin{array}{cc}x-1& \text{if }x\text{is even}\\ 2x& \text{if }x\text{is odd}\end{array}$

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