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

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:

The given rational expression is:

5x/x^2-25

The rational function is h(x)/g(x)

We can find the vertical asymptotes by solving the equation g(x) = 0

The zeroes of the function are the values of x when h(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.

According to the given fraction x^2-25 is g(x)

Lets solve it:

x^2-25 = 0

Add 25 at both sides:

x^2-25+25 = 0+25

x^2=25

Now take square root at both sides:

√x^2 = √25

x= +/- 5

The equations of the vertical asymptotes are:

x = -5 , x = 5

The degree of the numerator is less than the degree of the denominator because the degree of numerator is 1 and degree of denominator is 2.

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

Now to find the zero we have to make h(x)=0

where h(x)is the numerator.

5x=0

Divide both sides by 5:

5x/5 = 0/5

x= 0

The zero of the function is x = 0