Determine the domain and horizontal asymptote for the follow

Determine the domain and horizontal asymptote for the following functions. If there is no horizontal asymptote, potter, write DNE. Express the domain using interval notation. f (x) = x/4x - 7 Horizontal Asymptote y = g (x) = x^/z + 3 Domain Horizontal Asymptote y = h (x) = 4x + 7/3x + 12 Domain Horizontal Asymptote y =

Solution

(a) (x) = x/(4x-7). The domain of f(x) consists of all real numbers , except the one which makes the denominator 0. Thus, the domain of f(x) is ( -, 7/4) U (7/4, ). Further, since the numerator and the denominator have the same degree, the horizontal asymptote is y = ¼ (the ratio of the leading coefficients).

(b) g(x) = x²/(x+3). The domain of g(x) consists of all real numbers , except the one which makes the denominator 0. Thus, the domain of g(x) is ( -, -3) U (-3, ). Further, since the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote. DNE.

(c) h(x) = (4x+7)/(3x+12). The domain of h(x) consists of all real numbers , except the one which makes the denominator 0. Thus, the domain of h(x) is ( -, -4) U (-4, ). Further, since the numerator and the denominator have the same degree, the horizontal asymptote is y = 4/3(the ratio of the leading coefficients).