Final answer:
The function f(x) = (3x² - 33x + 84) / (5x² - 39x + 28) has a single vertical asymptote at x = 4/5 after factoring both the numerator and the denominator and simplifying the fraction.
Step-by-step explanation:
To find the vertical asymptotes of the function f(x) = (3x² - 33x + 84) / (5x² - 39x + 28), you need to determine the values of x that make the denominator equal to zero, as long as those x-values do not also make the numerator zero (which would indicate a hole rather than an asymptote).
First, factor the denominator:
5x² - 39x + 28 = (5x - 4)(x - 7)
Set each factor equal to zero to find the potential vertical asymptotes:
- 5x - 4 = 0 → x = 4/5
- x - 7 = 0 → x = 7
Next, factor the numerator and simplify the fraction if possible:
3x² - 33x + 84 = (3x - 12)(x - 7)
Now, we see that the term (x - 7) appears in both the numerator and the denominator, indicating a hole rather than a vertical asymptote at x = 7. Therefore, the only vertical asymptote is at x = 4/5.