Question

f(x)= (4x)/(x²-4)

Domain

Vertical Asymtote

Horizontal Asymtote

slant asymptote

Answer 1

1.f(x) =(4x)/(x²-4) is a rational function. The denominator, i.e. x²-4 has zeros at x = 2 and x = -2. Since division by 0 is not defined, the domain of f(x) is given by {x : x ∈ R, x ≠-2, x≠2}. In interval notation, the domain of f(x) is (-∞,-2)U (2,∞)

2. We know that when f(x) = p(x) / q(x), the equations of the vertical asymptotes can be found by finding the roots of q(x). Here, the roots of x²-4 ae -2 and 2. Hence the vertical asymptotes are x = -2 and x = 2.

3. The horizontal asymptote(s) is/are determined by looking at the degrees of the numerator (n) and denominator (m). If n<m, the X-axis i.e the line y = 0 is the horizontal asymptote. Here, the degrees of the numerator and the denominator are 1 and 2 respectively. Therefore, the X-axis i.e the line y = 0 is the horizontal asymptote.

4. If n = m+1, where n and m are as above, there is an oblique or slant asymptote. Since it is not the case here, there is no oblique or slant asymptote.