1. RATIONAL FUNCTIONS A rational function is a function of the form:
where p and q are polynomials

2. Domain
That’s because we’ve finally met a function with restrictions on x. We can’t take the log of a negative number because we can’t raise a number to a power and get a negative value.
For all other functions we’ve seen, x can be any real number.

3. The functions p and q are polynomials.
Defn:
Rational Function
The functions p and q are polynomials.
The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.

4. What would the domain of a rational function be?
We’d need to make sure the denominator  0
Find the domain.
We have a vertical asymptote at x=-3

5. What would the domain of a rational function be?
We’d need to make sure the denominator  0

6. What would the domain of a rational function be?
We’d need to make sure the denominator  0
If you can’t see it in your head, set the denominator = 0 and factor to find “illegal” values.

7. Let’s consider the graph
We recognize this function as the reciprocal function from our “library” of functions.
Can you see the vertical asymptote?
Let’s see why the graph looks like it does near 0 by putting in some numbers close to 0.
The closer to 0 you get for x (from positive direction), the larger the function value will be
Try some negatives

8. X-Intercepts are values of x that make the function = 0
We’d need to find values that make the numerator = 0
We have an x-intercept at (0, 0)

9. Does the function have an x intercept?
There is NOT a value that you can plug in for x that would make the function = 0. The graph approaches but never crosses the horizontal line y = 0. This is called a horizontal asymptote.
A graph will NEVER cross a vertical asymptote because the x value is “illegal” (would make the denominator 0)
A graph may cross a horizontal asymptote near the middle of the graph but will approach it when you move to the far right or left

10. X-Intercepts are values of x that make the function = 0
We’d need to find values that make the numerator = 0
We have an x-intercept at (3, 0)

11. X-Intercepts are values of x that make the function = 0
We’d need to find values that make the numerator = 0
We have an x-intercept at (1, 0).

12. So, what happens when we have the same factor in the numerator and in the denominator?
The graph will look like the simplified form of the function (when we cancel common factors), but with a restriction at that value.

13. Holes are created when there are common factors in the numerator and denominator.

14. Holes:
Holes:
X= -2
Or (-2, -10/3)
Vertical Asymptotes:
X= 1
X-Intercepts:
(0, 0), (3, 0)