Question

find the vertical asymtotes, x intercepts, horizontals asymptote, domain, and range

20) f(x) = -- x – 2 4.

Answer 1

20). We have f(x) = -4/(x-2) + 1 = (x-6)/(x-2).

It may be observed that f(x) is a rational function so that the equations of the vertical asymptote(s) can be found by finding the root(s) of the denominator.

Further, since 2 is the only root of the denominator (x-2), hence the line x = 2 is the only vertical asymptote of f(x).

Since the degrees of the numerator and denominator are same and since the ratio of the leading coefficients of the numerator and denominator is 1, hence the line y = 1 is the only horizontal asymptote.

Since the line x = 2 is the only vertical asymptote of f(x) ( also, since division by zero is not defined) , hence x can take all the real values other than 2. Therefore the domain of f(x) is (-∞,2)U(2,∞).

Similarly, since the line y = 1 is the only vertical asymptote of f(x) and since f(x) can take all the real values other than 1, hence the range of f(x) is (-∞,1)U(1,∞).

The x-intercept(s) is/are determined by substituting y = 0. Then x = 6 so that x = 6 is the only x-intercept.

A graph of f(x) is attached for visual clarity.

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