Question

which function is best represented by this graph?
https://flic.kr/p/2ov4nr4
select one:
a. $f(x) = \frac{5}{x+3}$
b. $f(x) = \frac{5}{x-3}$
c. $f(x) = \frac{x+5}{x+3}$
d. $f(x) = \frac{x+5}{x-3}$

Explanation:

Step1: Find vertical asymptote

The graph has a vertical asymptote at $x=-3$. For rational functions, vertical asymptotes occur where the denominator is 0. Set denominator to 0:
For option a: $x+3=0 \implies x=-3$
For option b: $x-3=0 \implies x=3$
For option c: $x+3=0 \implies x=-3$
For option d: $x-3=0 \implies x=3$
Eliminate options b and d.

Step2: Check horizontal asymptote

The graph approaches $y=0$ (x-axis) as $x\to\pm\infty$. For rational functions, if degree of numerator < degree of denominator, horizontal asymptote is $y=0$.
Option a: Numerator degree 0, denominator degree 1 → $y=0$
Option c: Numerator degree 1, denominator degree 1 → horizontal asymptote $y=\frac{1}{1}=1$, which does not match the graph.

Step3: Verify a point on graph

Pick $x=0$, the graph has $f(0)=\frac{5}{0+3}=\frac{5}{3}\approx1.67$, which matches the positive y-intercept shown.

Answer:

a. $f(x) = \frac{5}{x+3}$