Hello Loki 1.,
Let's run through the statements:
a) "g(x) is a rational function." By definition, a rational function is a quotient of two polynomial functions. The numerator (3x) and the denominator 4 - x^2 are both polynomial functions. Therefore, g(x) IS a rational function. TRUE
b) "g(x) is not defined when x = 0." Replace x by 0 in g(x); if it gives you a denominator equal to zero, then it is undefined and the statement is true:
g(0) = 3(0) / (4 - 0^2) = 0/4 = 0. The denominator was not zero, and we got an actual answer for g(0). So g(x) IS defined when x = 0. FALSE
c) "g(x) has one zero." Set the numerator = 0 and solve for x: 3x = 0 so x = 0. Only one zero which is x = 0. TRUE
d) "g(x) has two vertical asymptotes." Set the denominator = 0, solve for x. Any solutions will be your vertical asymptotes:
4 - x^2 = 0
---> 4 = x^2 so x = 2 or x = -2. Therefore, x = -2 and x = 2 are the vertical asymptotes of g(x). TRUE.
CONCLUSION: Statement (b) is False
