R(x)=x^2+x-6

________
x-3

1. find the domain and y-intercepts
2. find the x-intercepts
3. find the real solution of the equation
4. find the horizontal or oblique asymptotes
5. find the x-axis
6. plot the points

R(x) = (x^2 + x - 6)/(x-3)

= (x+3)(x-2)/(x-3)

1. domain: any real number, x ≠ 3
y-intercept, let x = 0, so y = 2
2. for x-intercepts, let y = 0
x = -3, 2
3. x = -3, y = 2
4. After a synthetic division, I got
R(x) = x+4 + 3/(x-3)
So the oblique asymptote is y = x+4

5. "find the x-axis" ????
6. you have to do that.

To answer your questions, let's go step by step:

1. Finding the domain and y-intercepts:
To find the domain of a rational function, we need to determine the values of x for which the function is defined. In this case, the rational function R(x) is defined for all real numbers except x = 3 (since that would result in division by zero).

To find the y-intercept, we can substitute x = 0 into the function and solve:
R(0) = (0^2 + 0 - 6) / (0 - 3) = -6 / -3 = 2
Therefore, the y-intercept is (0, 2).

2. Finding the x-intercepts:
To find the x-intercepts, we set the numerator equal to zero and solve for x:
x^2 + x - 6 = 0
This quadratic equation can be factored as (x - 2)(x + 3) = 0.
Setting each factor equal to zero, we get x - 2 = 0 and x + 3 = 0.
Solving these equations gives x = 2 and x = -3.
Therefore, the x-intercepts are (2, 0) and (-3, 0).

3. Finding the real solutions of the equation:
Since we already found the x-intercepts, the solutions are x = 2 and x = -3.

4. Finding the horizontal or oblique asymptotes:
To determine the horizontal asymptote, we look at the degree of the polynomial in the numerator and the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

To find the oblique asymptote, we use long division to divide the numerator by the denominator. Performing the division, we get:
x^2 + x - 6 divided by x - 3 = x + 4 + (6 / (x - 3))
This means that the oblique asymptote is y = x + 4.

5. Finding the x-axis:
The x-axis is simply the line where y = 0. In this case, we can see from the x-intercepts that the x-axis is crossed at x = 2 and x = -3.

6. Plotting the points:
We have the following points to plot:
- Y-intercept: (0, 2)
- X-intercepts: (2, 0) and (-3, 0)
- Oblique asymptote: y = x + 4
- X-axis crossings: (2, 0) and (-3, 0)

Now you can plot these points on a graph to visualize the rational function R(x).