Question

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

2. Is the function f (x)= 3x+ 4 from the set of integers to integers one to one? Why? 48​

Answer 1

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

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.