Question

Follow the seven step strategy to graph the following rational function.
f left parenthesis x right parenthesis equals StartFraction x plus 2 Over x squared plus x minus 20 EndFraction
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Part 1
To graph the​ function, first determine the symmetry of the graph of f. Choose the correct answer below.
neither​ y-axis symmetry nor origin symmetry
origin symmetry
​y-axis symmetry

Answer 1

To determine the symmetry of the rational function \( f(x) = \frac{x + 2}{x^2 + x - 20} \), we need to check for y-axis symmetry and origin symmetry.

1. **Y-axis Symmetry**: A function is symmetric with respect to the y-axis if \( f(-x) = f(x) \).

Let's compute \( f(-x) \):
\[
f(-x) = \frac{-x + 2}{(-x)^2 + (-x) - 20} = \frac{-x + 2}{x^2 - x - 20}
\]

Now compare \( f(-x) \) with \( f(x) \):
\[
f(x) = \frac{x + 2}{x^2 + x - 20}
\]
Since \( f(-x) \neq f(x) \), the function does not have y-axis symmetry.

2. **Origin Symmetry**: A function is symmetric with respect to the origin if \( f(-x) = -f(x) \).

Using \( f(-x) \) from above:
\[
f(-x) = \frac{-x + 2}{x^2 - x - 20}
\]
Now compute \( -f(x) \):
\[
-f(x) = -\frac{x + 2}{x^2 + x - 20} = \frac{-(x + 2)}{x^2 + x - 20} = \frac{-x - 2}{x^2 + x - 20}
\]

Compare \( f(-x) \) and \( -f(x) \):
\[
f(-x) = \frac{-x + 2}{x^2 - x - 20} \quad \text{and} \quad -f(x) = \frac{-x - 2}{x^2 + x - 20}
\]
Clearly, \( f(-x) \neq -f(x) \), so the function does not have origin symmetry.

Based on this analysis, the final answer is:

**Neither y-axis symmetry nor origin symmetry.**