Follow the seven step strategy to graph the following rational function.

Question

f left parenthesis x right parenthesisequalsStartFraction 4 x Over x minus 2 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.
​y-axis symmetry
origin symmetry
neither​ y-axis symmetry nor origin symmetry

Answer 1

To determine the symmetry of the rational function \( f(x) = \frac{4x}{x - 2} \), we can test for y-axis symmetry and origin symmetry.

1. Y-Axis Symmetry: A function is symmetric about the y-axis if \( f(-x) = f(x) \).

   • Calculate \( f(-x) \): \[ f(-x) = \frac{4(-x)}{-x - 2} = \frac{-4x}{-x - 2} = \frac{4x}{x + 2} \]
   • Since \( f(-x) = \frac{4x}{x + 2} \) is not equal to \( f(x) = \frac{4x}{x - 2} \), the function does not have y-axis symmetry.

2. Origin Symmetry: A function is symmetric about the origin if \( f(-x) = -f(x) \).

   • We already calculated \( f(-x) = \frac{4x}{x + 2} \).
   • Now, calculate \(-f(x)\): \[ -f(x) = -\left(\frac{4x}{x - 2}\right) = \frac{-4x}{x - 2} \]
   • Since \( f(-x) \neq -f(x) \), the function does not have origin symmetry.

Since the function does not exhibit y-axis symmetry or origin symmetry, the correct choice is:

Neither y-axis symmetry nor origin symmetry.