Describe a Linear Function: Key Identifiers and Characteristics

To solve this problem, we'll examine each expression to verify which represents a linear function:

• Option A: $x = y - 4$

This is a linear function. It can be written in the form $y = x + 4$, which matches the linear form $y = mx + c$ with $m = 1$ and $c = 4$.

• Option B: $x = 3x^2 + 1$

This equation involves a squared term ($x^2$), which means it's not a linear function. Linear functions do not have variables raised to powers other than one.

• Option C: $x = x + y - 1$

Rearrange to isolate $y$:

$y = 1$. This is a linear equation representing a horizontal line in the xy-plane.

• Option D: $x = x^2 + 4 - y$

This equation also involves a squared term ($x^2$), which disqualifies it as a linear function.

Based on this analysis, both Options A and C describe linear functions, and therefore the correct answer is that Answers A and C are correct.