Question

1. Determine whether the function f(x)= x³ from i to i is one to one. Explain.

Answer 1

Answer:

The function $f:\mathbb Z\to\mathbb Z,~f(x)=3x+4$ is injective (one-to-one).

Step-by-step explanation:

The definition of an injective function follows.

Let $X,Y$ be sets. Let $f:X\to Y$ be a function. We say $f$ is injective if, for all $x,y\in X$, $f(x)=f(y)$ implies $x=y$.

This is the proof that  $f:\mathbb Z\to\mathbb Z,~f(x)=3x+4$ is injective.

Let $x,y\in\mathbb Z$ and assume $f(x)=f(y)$. This means $3x+4=3y+4$. Subtracting $4$ gives $3x=3y$, then dividing by $3$ gives $x=y$. Thus $f$ is injective.