Question

Show all work to identify the asymptotes and zero of the function f of x equals 4 x over quantity x squared minus 16.

Answer 1

Zero: x=0

Asymptotes:

Horizontal: y=0

Vertical: x=4, x=-4

Explanation:

For the function

`f(x)=(4x)/(x^2-16)`

The degree of the numerator (1) is less than the degree of the denominator (2), so there is a horizontal asymptote at y=0.

The vertical asymptotes are found by setting the denominator equal to zero:

`x^2-16=0\\(x^2-16)+16=(0)+16\\x^2=16\\√(x^2)=√(16)`

`x=4` and
`x=-4`

The vertical asymptotes of the function are found at x=4 and x=-4

To find the zero of the function, set the numerator equal to zero:

`4x=0\\(4x)/(4)=(0)/(4)\\x=0`