Question
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Consider the following rational functions. \[ f(x)=\frac{3 x}{x^{2}-9} \] \( h(x)=\frac{3}{x^{2}-x-2} \) Choose the graph of each function from the choices

Ask by Collins Schmidt. in the United States
Mar 14,2025

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Answer

The graph of \( f(x) = \frac{3x}{x^2 - 9} \) has vertical asymptotes at \( x = -3 \) and \( x = 3 \), a horizontal asymptote at \( y = 0 \), passes through the origin, and is symmetric about the origin. The graph of \( h(x) = \frac{3}{x^2 - x - 2} \) has vertical asymptotes at \( x = -1 \) and \( x = 2 \), a horizontal asymptote at \( y = 0 \), a \( y \)-intercept at \( (0, -\frac{3}{2}) \), and no \( x \)-intercepts.

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The Deep Dive

To analyze the graphs of the functions \( f(x) \) and \( h(x) \), let's start with their characteristics. For \( f(x) = \frac{3x}{x^2 - 9} \), we notice that the denominator factors to \( (x-3)(x+3) \), indicating vertical asymptotes at \( x = 3 \) and \( x = -3 \). As \( x \) approaches these values, the function heads toward infinity or negative infinity, depending on the direction. Now, considering \( h(x) = \frac{3}{x^2 - x - 2} \), we can factor the denominator as \( (x-2)(x+1) \), leading to vertical asymptotes at \( x = 2 \) and \( x = -1 \). The function will also have horizontal asymptotes that we can determine by looking at the degrees of the polynomial in the numerator and denominator. So, when you choose the graphs, look for the key features: vertical asymptotes for both functions and their general behavior as \( x \) approaches those asymptotes! Happy graph hunting!

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