Determine the equation for the vertical asymptotes, if they exist, for each function. Then, state the domain.
f(x)= -1/x^2-7x+6
Assuming you meant:
f(x)= -1/(x^2-7x+6)
= -1/((x-6)(x-1))
vertical asymptotes are caused by the denominator being zero,
so x-6 = 0 ----> x = 6
or
x-1 = 0 ---> x = 1
To determine the equation for the vertical asymptotes, we need to find where the function is undefined. In other words, we need to find the values of x that make the denominator zero, because dividing by zero is undefined.
The denominator of the function is x^2 - 7x + 6. To find where this is equal to zero, we can set the denominator equal to zero and solve for x:
x^2 - 7x + 6 = 0
This equation can be factored as:
(x - 1)(x - 6) = 0
Setting each factor equal to zero gives us:
x - 1 = 0 or x - 6 = 0
Solving for x in each case, we have:
x = 1 or x = 6
So, the vertical asymptotes are x = 1 and x = 6. These are the values that make the denominator zero.
Next, let's state the domain of the function. The domain is the set of all possible input values of x for which the function is defined. In this case, the function is defined for all values of x except the values that make the denominator zero (which would result in division by zero).
Therefore, the domain of the function f(x) = -1/(x^2 - 7x + 6) is all real numbers except x = 1 and x = 6. In interval notation, we could express the domain as (-∞, 1) U (1, 6) U (6, ∞).