To determine the coordinates of the point of intersection of the asymptotes for the function [tex]\( r(x) \)[/tex], we first need to understand the asymptotes of the original function [tex]\( f(x) \)[/tex] and how they transform when defining [tex]\( r(x) \)[/tex].
### Given Function
The original function is given by:
[tex]\[ f(x) = \frac{2}{x-1} - 2 \][/tex]
#### Step 1: Vertical Asymptote of [tex]\( f(x) \)[/tex]
To find the vertical asymptote, we look for values of [tex]\( x \)[/tex] that make the denominator zero:
[tex]\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \][/tex]
Thus, [tex]\( x = 1 \)[/tex] is the vertical asymptote of [tex]\( f(x) \)[/tex].
#### Step 2: Horizontal Asymptote of [tex]\( f(x) \)[/tex]
To find the horizontal asymptote, we analyze the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex]:
[tex]\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \left( \frac{2}{x-1} - 2 \right) = 0 - 2 = -2 \][/tex]
Thus, [tex]\( y = -2 \)[/tex] is the horizontal asymptote of [tex]\( f(x) \)[/tex].
### Transformed Function [tex]\( r(x) \)[/tex]
The function [tex]\( r(x) \)[/tex] is defined as:
[tex]\[ r(x) = f(x-3) + 7 \][/tex]
This involves two transformations:
1. Horizontal shift by 3 units to the right.
2. Vertical shift upwards by 7 units.
#### Step 3: Vertical Asymptote of [tex]\( r(x) \)[/tex]
Starting with the vertical asymptote of [tex]\( f(x) \)[/tex], which is [tex]\( x = 1 \)[/tex], shifting this by 3 units to the right:
[tex]\[ x = 1 + 3 = 4 \][/tex]
Thus, [tex]\( x = 4 \)[/tex] is the vertical asymptote of [tex]\( r(x) \)[/tex].
#### Step 4: Horizontal Asymptote of [tex]\( r(x) \)[/tex]
Starting with the horizontal asymptote of [tex]\( f(x) \)[/tex], which is [tex]\( y = -2 \)[/tex], shifting this up by 7 units:
[tex]\[ y = -2 + 7 = 5 \][/tex]
Thus, [tex]\( y = 5 \)[/tex] is the horizontal asymptote of [tex]\( r(x) \)[/tex].
### Step 5: Intersection of the Asymptotes
The point of intersection of the asymptotes is where the vertical asymptote [tex]\( x = 4 \)[/tex] intersects the horizontal asymptote [tex]\( y = 5 \)[/tex].
Thus, the coordinates of the point of intersection of the asymptotes for [tex]\( r(x) \)[/tex] are
[tex]\[ (4, 5) \][/tex]
