Graph the rational function

Question

Fofx=2x+2/x+4

Answer 1

To graph the rational function f(x) = (2x + 2)/(x + 4), we need to follow these steps:

1. Determine the vertical asymptote(s): Set the denominator equal to zero and solve for x.
x + 4 = 0
x = -4
Therefore, the vertical asymptote is x = -4.

2. Determine the horizontal asymptote: If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote. In this case, the degrees are the same (both 1), so there is a horizontal asymptote at y = 2/1 = 2.

3. Determine the x-intercept(s): Set the numerator equal to zero and solve for x.
2x + 2 = 0
x = -1
Therefore, the x-intercept is (-1, 0).

4. Determine the y-intercept: Plug in x = 0 into the function.
f(0) = (2(0) + 2)/(0 + 4) = 2/4 = 1/2
Therefore, the y-intercept is (0, 1/2).

Now, let's plot these points and identify the behavior near the asymptotes.

As x approaches -4 from the left side (x < -4), f(x) becomes infinitely large (positive infinity).
As x approaches -4 from the right side (x > -4), f(x) becomes infinitely large (negative infinity).

Here is the graph of the rational function f(x) = (2x + 2)/(x + 4):

Graph:
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-4 -1 0

(Note: The graph does not include the vertical asymptote at x = -4 because the program cannot display a vertical line.)