describe the vertical asymptote(s) and hole*s) for the graph of y=(x+2)(x+4)/(x+4)(x+1)
The function y=(x+2)(x+4)/(x+4)(x+1) has one vertical asymptote at x=-4, where the denominator becomes zero. This vertical asymptote arises because the factor (x+4) in the denominator cancels out with the factor (x+4) in the numerator.
The function also has a hole at x=-1, where the factor (x+1) in the denominator cancels out with the factor (x+1) in the numerator. This cancellation results in a hole instead of a vertical asymptote because the factors completely cancel out at that point.
a asymptotes x=-1 and hole x=4
b asymptote x=2 and hole x=-1
c asymptote x=-1 and hole x=-4
d asymptote x=1 and hole x=4
The correct answer is c) asymptote x=-1 and hole x=-4
Explanation:
- The function y=(x+2)(x+4)/(x+4)(x+1) has a vertical asymptote at x=-1, where the factor (x+1) in the denominator cancels out with the factor (x+1) in the numerator.
- The function also has a hole at x=-4, where the factor (x+4) in the denominator cancels out with the factor (x+4) in the numerator. This cancellation results in a hole instead of a vertical asymptote because the factors completely cancel out at that point.