Question

identify the graph of $f(x)=\frac{10 - 10x -}{x^2}$ ?

Explanation:

Step1: Analyze the function

The function \( f(x)=\frac{10 - 10x}{x^{2}}=\frac{10}{x^{2}}-\frac{10}{x}\).

Step2: Find the vertical asymptote

Set the denominator \(x^{2}=0\), we get \(x = 0\) as the vertical asymptote.

Step3: Analyze the behavior as \(x\to\pm\infty\)

As \(x\to\pm\infty\), \(f(x)=\frac{10 - 10x}{x^{2}}=\frac{10}{x^{2}}-\frac{10}{x}\to0\). So \(y = 0\) is the horizontal asymptote.

Step4: Analyze the \(x\)-intercept

Set \(y = 0\), then \(\frac{10-10x}{x^{2}}=0\), \(10-10x = 0\), \(x = 1\).

Step5: Analyze the \(y\)-intercept

The function \(f(x)\) is not defined at \(x = 0\), so there is no \(y\)-intercept.

By analyzing the asymptotes, intercepts and the general form of the rational function, we can match the graph.

Answer:

The first graph (the one with \(x = 0\) as vertical asymptote, \(y=0\) as horizontal asymptote and \(x\)-intercept at \(x = 1\)) is the graph of \(y=\frac{10 - 10x}{x^{2}}\).