To find the vertical asymptotes of the function f(x) = 10 / (x² – 7x – 30), we first need to determine where the denominator equals zero since vertical asymptotes occur at these points.
We begin by solving the equation:
x² – 7x – 30 = 0
Next, we can factor the quadratic. We are looking for two numbers that multiply to -30 and add up to -7. The numbers -10 and 3 work:
(x – 10)(x + 3) = 0
Setting each factor equal to zero gives us:
x – 10 = 0 or x + 3 = 0
Solving these equations, we find:
x = 10 and x = -3
These x-values indicate the locations of the vertical asymptotes of the function. Therefore, the vertical asymptotes of f(x) are:
x = 10 and x = -3
This means that as the function approaches these x-values, the value of f(x) will approach infinity or negative infinity, indicating the presence of vertical asymptotes at those points.