\( y=\frac{3}{x}+3 \) and \( 3 y-x \)

The equation \( y = \frac{3}{x} + 3 \) represents a hyperbola that has an asymptote at \( y = 3 \) and approaches this line as \( x \) increases or decreases. Interestingly, as \( x \) gets larger or smaller in magnitude, \( y \) approaches 3, but never actually touches it, making it a great example of how limits work in calculus! On the other hand, \( 3y - x \) can represent a linear relationship between \(x\) and \(y\) when expressed in slope-intercept form. If you rearrange it to isolate \(y\), you’ll find a straight line where the slope indicates how much \(y\) changes with respect to \(x\). It's a fun challenge to identify where this line intersects with the hyperbola, leading to a deeper understanding of the interplay between linear and nonlinear functions!