Mathematics

Question

What is the relationship between the range and the equation of the asymptote

1 Answer

  • Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. (They can also arise in other contexts, such as logarithms, but you'll almost certainly first encounter asymptotes in the context of rationals.)  I'll give you an example:


     

    This is a rational function. More to the point, this is a fraction. Can you have a zero in the denominator of a fraction? No. So if I set the denominator of the above fraction equal to zero and solve, this will tell me the values that x cannot be:

    x2 – 5x – 6 = 0
    (x – 6)(x + 1) = 0
    x = 6 or –1

    So x cannot be 6 or –1, because then I'd be dividing by zero.


     

    The domain is the set of all x-values that I'm allowed to use. The only values that could be disallowed are those that give me a zero in the denominator. So I'll set the denominator equal to zero and solve.

    x2 + 2x – 8 = 0
    (x + 4)(x – 2) = 0
    x = –4 or x = 2

    Since I can't have a zero in the denominator, then I can't have x = –4 or x = 2 in the domain. This tells me that the vertical asymptotes (which tell me where the graph can not go) will be at the values x = –4 or x = 2.

    domain: 
    vertical asymptotes:  x = –4, 2  


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