In this function, f(x)= [2/(x-3)]+1: I know it is a rational function and the graph moves 1 unit up and 3 units right. I also see that the 2 makes the function move closer, but I don't understand why it does that in mathematical terms.

I do not understand your statement:

"2 makes the function move closer" closer to what?

1 is the asymptotic value of f(x) for very large postive or large negative x. The function goes to infinity at x=3, because of th4 zero denominator there.

thank you for replying, but i figured it out

To understand how the constant term in a rational function affects the position of the graph, let's analyze the given function:

f(x) = 2/(x - 3) + 1

We can see that the constant term in this case is "+1". To visualize the effect of this term on the graph, let’s consider the behavior of the function when x is very large or very small.

When x becomes infinitely large (approaching positive or negative infinity), the "+1" becomes less significant compared to the rest of the expression. Consequently, the function behaves similarly to f(x) = 2/(x - 3). This implies that the graph of the function will have an asymptote at x = 3, where the function approaches positive or negative infinity. This asymptote occurs because the denominator (x - 3) approaches zero as x approaches 3, resulting in a vertical asymptote.

On the other hand, when x becomes infinitely small, the "+1" dominates the rest of the expression since it is a constant term. This means that the function behaves similarly to f(x) = 1. Consequently, the graph intersects the y-axis at y = 1.

So, in summary, the constant term "+1" in the function f(x) = 2/(x - 3) + 1 mainly affects the vertical position of the graph. It shifts the graph upward by 1 unit, causing it to intersect the y-axis at y = 1.