Graph and label the following two functions: f(x)=(x^2+7x+12)/(x+4) g(x)=(-x^2+3x+9)/(x-1) 1. Describe the domain and range for each of these functions. 2. Determine the equation(s) of any asymptotes found in the graphs of these functions, showing all work. 3. Discuss the conditions needed for a rational function to contain a slant asymptote.

f is any x except x=-4, g is any x except x=1

range, consider f
f when x=-inf ; f= (inf)^2/inf
when x=+inf; f= (inf)^2/inf

so range of f is -inf to inf

consider g
when x=-inf; g-inf^2/-inf= -inf
x=+inf; g= inf^2/inf= inf

range is -inf to inf

To graph and label the functions f(x) and g(x), we can follow these steps:

Step 1: Find the x-intercepts (where the functions intersect the x-axis) by setting each function equal to zero and solving for x.

For f(x):
(x^2 + 7x + 12) / (x + 4) = 0
(x + 3)(x + 4) / (x + 4) = 0
(x + 3) = 0

So, the x-intercept for f(x) is x = -3.

For g(x):
(-x^2 + 3x + 9) / (x - 1) = 0
(-1)(x - 3)(x + 3) / (x - 1) = 0
(x - 3)(x + 3) = 0
(x - 3) = 0 or (x + 3) = 0

So, the x-intercepts for g(x) are x = 3 and x = -3.

Step 2: Find the y-intercept (where the functions intersect the y-axis) by plugging in x = 0 into each function.

For f(x):
f(0) = (0^2 + 7(0) + 12) / (0 + 4)
f(0) = 12/4
f(0) = 3

So, the y-intercept for f(x) is y = 3.

For g(x):
g(0) = (-0^2 + 3(0) + 9) / (0 - 1)
g(0) = 9/-1
g(0) = -9

So, the y-intercept for g(x) is y = -9.

Step 3: Determine the vertical asymptotes (lines that the graph approaches but never touches) by finding the values of x that make the denominator of each function equal to zero but not the numerator.

For f(x):
x + 4 = 0
x = -4

So, there is a vertical asymptote at x = -4 for f(x).

For g(x):
x - 1 = 0
x = 1

So, there is a vertical asymptote at x = 1 for g(x).

Step 4: Determine the horizontal asymptotes (lines that the graph approaches as x approaches positive or negative infinity) by comparing the degrees of the numerator and denominator.

For f(x):
The degree of the numerator (2) is equal to the degree of the denominator (1).

In this case, there is no horizontal asymptote for f(x). Instead, f(x) has a slant asymptote.

Step 5: Determine the slant asymptote (a linear function) by dividing the numerator by the denominator using polynomial long division.

For f(x):
(x^2 + 7x + 12) / (x + 4) = x + 3 + (-24) / (x + 4)
f(x) ≈ x + 3 as x approaches positive or negative infinity.

So, the slant asymptote for f(x) is y = x + 3.

For g(x):
The degree of the numerator (2) is greater than the degree of the denominator (1), so there is no slant asymptote for g(x).

Now, let's discuss the domain and range for each function:

Domain: The domain of a rational function is all real numbers except for the values that make the denominator equal to zero, leading to undefined values.

For f(x), the denominator is (x + 4), so x cannot equal -4. Therefore, the domain of f(x) is all real numbers except x = -4.

For g(x), the denominator is (x - 1), so x cannot equal 1. Therefore, the domain of g(x) is all real numbers except x = 1.

Range: The range of a rational function depends on the behavior of the function as x approaches positive or negative infinity.

For f(x), the slant asymptote is y = x + 3. As x approaches positive or negative infinity, the function f(x) gets arbitrarily close to the slant asymptote. Therefore, the range of f(x) is all real numbers.

For g(x), there is no slant asymptote or horizontal asymptote. As x approaches positive or negative infinity, g(x) also gets arbitrarily close to positive or negative infinity. Therefore, the range of g(x) is all real numbers.

In summary:
1. The domain of f(x) is all real numbers except x = -4, and the range is all real numbers.
The domain of g(x) is all real numbers except x = 1, and the range is all real numbers.

2. The vertical asymptote for f(x) is x = -4, and for g(x) is x = 1.
f(x) has a slant asymptote at y = x + 3, while g(x) does not have any asymptotes.

3. A rational function will contain a slant asymptote if the degree of the numerator is one greater than the degree of the denominator. In other words, if the function is a proper fraction (degree of numerator < degree of denominator).