Show all work to identify the asymptotes and zero of the function f of x equals 3 x over quantity x squared minus 9.

The equations of the vertical asymptotes are x = -3 , x = 3

The horizontal asymptote is the x-axis of equation y = 0

The zero of the function is x = 0

Step-by-step explanation:

* Lets explain how to solve the problem

- The rational function is f(x)/g(x) , where f(x) is the numerator of it and

  g(x) is its denominator

- Vertical asymptotes can be found by solving the equation g(x) = 0

- If the numerator f(x) is a lower degree than the denominator g(x),

 then the x-axis (y = 0) is the horizontal asymptote.

- The zeroes of the function are the values of x when f(x) = 0

* lets solve the problem

∵ The rational function

- The numerator is 3x and the denominator is x² - 9

∵ Vertical asymptotes can be found by solving the equation x² - 9 = 0

∵ x² - 9 = 0 ⇒ add 9 to both sides

∴ x² = 9 ⇒ take √ for both sides

∴ x = ± 3

∴ The equations of the vertical asymptotes are x = -3 , x = 3

∵ The degree of the numerator is 1

∵ The degree of the denominator is 2

∴ The degree of the numerator < the degree of the denominator

∴ The horizontal asymptote is the x-axis of equation y = 0

∵ To find the zero of the function put the numerator = 0

∵ The numerator is 3x

∴ 3x = 0 ⇒ divide both sides by 3

∴ x = 0

∴ The zero of the function is x = 0

# Look to the attached graph for more understanding

x=6 and x=3

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step-by-step explanation: