Answer:
The function [tex]f:\mathbb Z\to\mathbb Z,~f(x)=3x+4[/tex] is injective (one-to-one).
Step-by-step explanation:
The definition of an injective function follows.
Let [tex]X,Y[/tex] be sets. Let [tex]f:X\to Y[/tex] be a function. We say [tex]f[/tex] is injective if, for all [tex]x,y\in X[/tex], [tex]f(x)=f(y)[/tex] implies [tex]x=y[/tex].
This is the proof that [tex]f:\mathbb Z\to\mathbb Z,~f(x)=3x+4[/tex] is injective.
Let [tex]x,y\in\mathbb Z[/tex] and assume [tex]f(x)=f(y)[/tex]. This means [tex]3x+4=3y+4[/tex]. Subtracting [tex]4[/tex] gives [tex]3x=3y[/tex], then dividing by [tex]3[/tex] gives [tex]x=y[/tex]. Thus [tex]f[/tex] is injective.
