You can put this solution on YOUR website!
First, please don't post in all caps. It is hard to read and it is considered shouting.
Second, include the entire problem. (x+3x)/x is just an expression. A function should be written as an equation. Something like
y = (x+3x)/x
f(x) = (x+3x)/x
Third, is it really (x+3x)/x? The numerator simplifies to 4x. Usually functions are defined without such obvious simplifications.
Fourth, if the function really is y = (x+3x)/x then, contrary to what you determined, there are no intercepts and there are no asymptotes for this function. I'll explain why below.
When analyzing rational functions, you should start by looking for "holes". x values that make both the numerator and denominator zero, if any, will be a "hole" in the graph. We look for holes first because look for if there are some, they affect how we find intercepts and asymptotes.
For our function,
y = 4x/x
It is quite simple to see that if x = 0, then both the numerator and denominator will be zero. (With more complex functions you may not be able to see the holes. For these you should factor the numerator and denominator.) So there will be a hole at x = 0. (x = 0 is also where y-intercepts would be. But we will have a hole there instead of a y-intercept!)
When you find a hole:
- Remember them! Only holes will occur for those x values.
- Factor the function and cancel common factors.
- Graph the reduced function ... putting little holes in the graph of the reduced function for the appropriate x values.
Let's see this your function:
1. We have one hole, at x = 0
2. Reduce the function.
The x's cancel leaving:
y = 4
3. Graph the reduced function.
The equation y = 4 is not a rational function. It is the equation of a horizontal line where the y coordinates are 4's ... except we will have a hole in this line at x = 0 (the y-axis). A hole is represented by a small, open circle (a small "oh").
In summary, the graph of y = 4x/x is a horizontal line with y-coordinates of 4 and with a hole at x = 0 (the y=axis). It has no x or y intercepts (the hole does not count as an intercept) and there are no asymptotes.