Question

choose a possible sketch of the graph of ( y = \frac{4}{x} ) and ( y = \frac{x - 1}{x + 3} ) to demonstrate that there are two points of intersection.
a.
graph a
b.
graph b
c.
graph c
d.
graph d

Explanation:

Step1: Identify first function

$y=\frac{1}{x}$ is a hyperbola with branches in Q1/Q3, asymptotes $x=0,y=0$.

Step2: Identify second function

Rewrite $y=\frac{x-1}{x+3}=1-\frac{4}{x+3}$. This is a hyperbola shifted left 3, up 1, asymptotes $x=-3,y=1$.

Step3: Find intersection points

Set $\frac{1}{x}=1-\frac{4}{x+3}$. Cross multiply:
$x+3 = x(x+3)-4x$
$x+3 = x^2+3x-4x$
$x^2-2x-3=0$
Factor: $(x-3)(x+1)=0$, so $x=3,x=-1$. Two real solutions, meaning two intersection points.

Step4: Match to graph

Only option A shows two distinct intersection points between the two curves.

Answer:

A. <The graph with two distinct intersection points between the two curves>