Show all work to identify the asymptotes and zero of the function f of x equals 5 x over quantity x squared minus 25.

The equations of the vertical asymptotes are x = -5 , x = 5

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 h(x)/g(x) , where h(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 h(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 h(x) = 0

* Lets solve the problem

∵ The rational function is

- The numerator is 5x and the denominator is x² - 25

∵ Vertical asymptotes can be found by solving the equation x² - 25 = 0

∵ x² - 25 = 0 ⇒ add 15 to both sides

∴ x² = 25 ⇒ take √ for both sides

∴ x = ± 5

∴ The equations of the vertical asymptotes are x = -5 , x = 5

∵ 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 5x

∴ 5x = 0 ⇒ divide both sides by 5

∴ x = 0

∴ The zero of the function is x = 0

# The attached graph for more understanding