To identify which graph represents the rational function \( f(x) = \frac{x^2 + 3x + 2}{2x^2 - x - 2} \), we can analyze the function by factoring and determining its characteristics such as vertical and horizontal asymptotes, intercepts, and behavior.
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Factoring the function:
- The numerator can be factored as \( (x + 1)(x + 2) \).
- The denominator can be factored as \( (2x + 4)(x - 1) \) or rewritten directly to see its roots.
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Finding asymptotes:
- Vertical Asymptotes: Set the denominator to zero: \( 2x^2 - x - 2 = 0 \). The roots are \( x = 2 \) and \( x = -1 \). Thus, \( x = 2 \) is a vertical asymptote.
- Horizontal Asymptote: Since the degree of the numerator is equal to that of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients, which is \( y = \frac{1}{2} \).
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Intercepts:
- X-Intercepts: Set the numerator to zero: \( x^2 + 3x + 2 = 0 \). This gives intercepts at \( x = -1 \) and \( x = -2 \).
- Y-Intercept: Set \( x = 0 \) in \( f(x) \) giving \( f(0) = \frac{2}{-2} = -1 \).
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Behavior near the asymptotes:
- As \( x \) approaches \( 2 \) from the left, \( f(x) \) approaches \( \infty \).
- As \( x \) approaches \( 2 \) from the right, \( f(x) \) approaches \( -\infty \).
- The function approaches the horizontal asymptote \( y = \frac{1}{2} \) as \( x \to \pm \infty \).
Now we can compare these results with the provided options.
- Option Descriptions:
- Both Option 1 and Option 2: They mention a vertical dashed line at \( x = 2 \) and a horizontal line at \( y = 1 \), which cannot represent the function correctly since we established that the horizontal asymptote is \( y = \frac{1}{2} \).
- Option 3: Again, it mentions the horizontal line at \( y = 1 \).
- Option 4: It indicates a vertical asymptote at \( x = 2 \) and mentions a horizontal line, although it inaccurately states that the horizontal asymptote is at another value.
None of the options perfectly match the horizontal asymptote at \( y = \frac{1}{2} \). Therefore, while all options provide important details regarding vertical asymptotes, their incorrect identification of the horizontal asymptote leads to an incorrect representation of the function.
To summarize, if we had to pick the one closest to our findings, we would say Option 4 is the best among poor choices, since it acknowledges the vertical asymptote at \( x = 2 \) (but it still fails on the horizontal asymptote details). Thus, great caution is advised when choosing based on the mistakes in presented options.
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