g(x) = x^2 - x -12 / x + 1
I know the division brings it to x -2 + -10/x + 1
but these are the steps I have to follow
1. factor the numerator and denominator and find its domain.
( I know 0 is not in domain)
2. write in lowest ermas as p(x)/ q(x) and find the real zeros of the numerator that is find the ral solutions of the equation p(x) = 0
3. find the ral zeros of the denominator that is find the real solution of q(x) = 0
4. locate horizontal or oblique asymptotes determine points at which the graph intersects these asymptotes
5. using the real zeros of the numerator and denominator divide the x axis into intervals and determine where the graph is above the x axis and where below the x axis
6. analyze the behavior of the graph near each asymptote and indicate behavior on the graph.
7. graph ( this I can do )
Help me! This woman is impossible!!!!!
What woman? It's just a math problem, like others in your book...
You have (x-4)(x+3)/(x+1)
domain is all reals except where denominator is zero. That is all reals except x = -1
zeros at x = 4 and -3 because that's where the numerator is zero and the denominator is not zero
no horizontal asymptotes, since the numerator has higher degree than the denominator. As x gets large, the fraction is just x^2/x = x
So, oblique asymptote is y=x
The graph intersects it at (-6,-6)
A little interval chart will show that
y<0 for x in (-oo,-3)U(-1,4)
y>0 for x in (-3,-1)U(4,oo)
At the end you say you can graph it. Well! In that case, what's the problem? The graph shows all the answers to the questions.
I'm sorry to hear that you're having trouble with this problem. I'll do my best to walk you through the steps.
1. To factor the numerator and denominator and find their domain:
a. Factor the numerator: g(x) = (x^2 - x - 12) / (x + 1)
The numerator can be factored as (x - 4)(x + 3).
b. Factor the denominator: (x + 1)
c. The domain is the set of all real numbers except for those that make the denominator equal to zero. So, x cannot be equal to -1.
2. Write in lowest terms and find the real zeros of the numerator:
After factoring the numerator, we can write g(x) as g(x) = (x - 4)(x + 3) / (x + 1).
To find the real zeros, set the numerator equal to zero and solve for x:
(x - 4)(x + 3) = 0
Setting each factor equal to zero, we have x - 4 = 0 and x + 3 = 0.
Solving these equations, we find x = 4 and x = -3 are the real zeros of the numerator.
3. Find the real zeros of the denominator:
We already found the real zero of the denominator in step 1, which is x = -1.
4. Locate horizontal or oblique asymptotes and determine points of intersection:
To find horizontal asymptotes, we compare the degrees of the numerator and denominator.
Since the degree of the numerator is 2 and the degree of the denominator is 1, there is no horizontal asymptote.
Oblique asymptotes can be found using long division or synthetic division, but in this case, we don't have one.
5. Determine where the graph is above or below the x-axis:
Choose test points from each of the intervals created by the zeros of the function. By plugging in these test points into the function, we can determine if the function is positive or negative for each interval. This will tell us whether the graph is above or below the x-axis in each interval.
6. Analyze the behavior of the graph near each asymptote and indicate it on the graph:
Since we don't have any asymptotes in this case, we don't need to analyze the behavior near them.
7. Graph the function:
Finally, use all the information gathered from the previous steps to graph the function. Plot the points, determine the shape of the graph based on the zeros and asymptotes, and show the behavior of the graph near any intercepts or asymptotes identified.
I hope this helps! Let me know if you have any more questions.