We are tasked with analyzing the function $g: \mathbb{Z} \to \mathbb{Z}$, defined as:

$g(x) = x - 7$

We need to determine whether the function is onto (surjective), one-to-one (injective), or bijective (both injective and surjective). Let’s evaluate step by step.

1. Onto (Surjective):

A function is surjective if every possible output in the codomain $\mathbb{Z}$ (set of integers) has at least one input $x$ in the domain $\mathbb{Z}$.

• The function $g(x) = x - 7$ produces outputs that are all integers because for any integer $y \in \mathbb{Z}$, we can solve for $x$ such that: $g(x) = y \implies x - 7 = y \implies x = y + 7.$ Since $x = y + 7$ is also an integer (as $y$ is an integer and adding 7 preserves the integer property), every integer $y$ in the codomain $\mathbb{Z}$ has a pre-image.

Conclusion: The function is onto (surjective).

2. One-to-One (Injective):

A function is injective if each output corresponds to exactly one input.

To test injectivity, assume $g(x_1) = g(x_2)$. Then:

$x_1 - 7 = x_2 - 7.$

Simplifying:

$x_1 = x_2.$

This shows that the function is injective, as equal outputs imply equal inputs.

Conclusion: The function is one-to-one (injective).

3. Bijective:

A function is bijective if it is both injective and surjective.

From our earlier conclusions:

• The function is surjective (onto).
• The function is injective (one-to-one).

Therefore, $g(x) = x - 7$ is bijective.

Final Answers:

• ONTO (surjective): Yes
• ONE-TO-ONE (injective): Yes
• BIJECTIVE: Yes

Questions to expand your understanding:

1. What does it mean for a function to be surjective?
2. How do you test if a function is injective?
3. Can a function be surjective but not injective? Give an example.
4. Can a function be injective but not surjective? Give an example.
5. What changes if the codomain is not the same as the range of the function?

Tip: Always check injectivity and surjectivity independently when determining if a function is bijective.