Question

which of the following is the expression of the graph of the rational function below?
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Explanation:

Step1: Identify Vertical Asymptotes

From the graph, vertical asymptotes are at \( x = 0 \) and \( x = 3 \) (assuming grid lines, need to check the graph's vertical dashed lines). So the denominator should have factors \( x(x - 3) \), so denominator is \( x^2 - 3x \).

Step2: Identify Horizontal Asymptote

The horizontal asymptote seems to be \( y = 0 \) (since the graph approaches a horizontal line, likely degree of numerator < degree of denominator).

Step3: Identify x-intercept

The x-intercept is at \( x = - 4 \) (from the graph, where it crosses the x-axis). So numerator should have a factor \( (x + 4) \), so numerator is \( x + 4 \).

Step4: Form the Rational Function

Putting it together, the function could be \( f(x)=\frac{x + 4}{x(x - 3)}=\frac{x + 4}{x^2 - 3x} \). We can check the behavior: vertical asymptotes at \( x = 0 \) and \( x = 3 \), x-intercept at \( x=-4 \), horizontal asymptote \( y = 0 \) (since degree of numerator (1) < degree of denominator (2)).

Answer:

\( \boldsymbol{f(x)=\frac{x + 4}{x^2 - 3x}} \) (or equivalent simplified form, depending on the options. Since options are not provided, this is the derived function from the graph's features: vertical asymptotes at \( x = 0, x = 3 \), x-intercept at \( x=-4 \), horizontal asymptote \( y = 0 \)).