f(x) = (x^3+5x^2-28x-32)/(x^3-18x^2+105x-196)
which can be written as
f(x) = [(x+1)(x-4)(x+8)]/[(x-7)^2(x-4)
The domain of the function f(x), in interval notation from left to right, is ____ U ____ U ______.
The root(s) of f(x), in increasing order, is/are: _____, ____.
f(x) has hole(s) at the point(s): (____,____).
f(x) has vertical asymptotes when x is: ___.
f(x) has a horizontal asymptote at y = ___.
The domain of the function is all Real, except for those two points which will result in a zero in the denominator, and hence f(x) will not be defined. One of these points is x=7.
The roots can be deduced from the numerator of f(x). One of the roots is x+1=0, or x=-1.
Vertical asymptotes occur where the denominator become zero.
Horizontal asymptote is the limit of the function when x->-&inf; or x->+&inf;.
In this case, both converge to y=1.
Vertical asymptote at x = 7 , and a "hole" at (4, 20/3)
To find the domain of the function f(x), we need to consider any values of x that would make the denominator equal to zero, since division by zero is undefined. In this case, the denominator is given by (x - 7)^2(x - 4). Therefore, the function is undefined for x = 7 (double root) and x = 4. Any other value of x is within the domain.
So, the domain of the function f(x) is (-∞, 4) U (4, 7) U (7, ∞) in interval notation from left to right.
To find the root(s) of f(x), we need to solve the numerator (x + 1)(x - 4)(x + 8) equal to zero. This means x + 1 = 0 or x - 4 = 0 or x + 8 = 0. We find that x = -1, x = 4, and x = -8 are the roots of the function f(x).
Therefore, the root(s) of f(x), in increasing order, is/are: -8, -1, 4.
A hole exists in the graph of f(x) when there is a common factor between the numerator and denominator that can be canceled out. In this case, the factor (x - 4) appears in both the numerator and denominator, so there is a hole at x = 4.
Therefore, the hole(s) in f(x) is/are: (4, f(4)).
To find the vertical asymptotes of f(x), we look for the values of x that make the denominator equal to zero, but are not canceled out by any factors in the numerator. In this case, x = 7 is a double root, so there is a vertical asymptote at x = 7.
Therefore, f(x) has vertical asymptotes when x is 7.
To find the horizontal asymptote of f(x), we need to examine the behavior of the function as x approaches positive or negative infinity. As we observe the degree of the numerator and denominator, we can see that the highest power of x in both is x^3. Hence, the quotient of their coefficients will give us the horizontal asymptote.
Therefore, the horizontal asymptote of f(x) is y = 1.
In summary:
The domain of the function f(x), in interval notation from left to right, is (-∞, 4) U (4, 7) U (7, ∞).
The root(s) of f(x), in increasing order, is/are: -8, -1, 4.
f(x) has hole(s) at the point(s): (4, f(4)).
f(x) has vertical asymptotes when x is 7.
f(x) has a horizontal asymptote at y = 1.