Name
November 12(G), or 18(H), 2014
Math 4 notes and problems
page 1
Rational functions with removable discontinuities
Objective: Graph rational functions that have removable discontinuities.
Definitions and concepts
Rational function
A rational function is a function of the form gf (( xx)) where f(x) and g(x) are polynomials.
Note that any zeros of g(x) must be excluded from the domain of the rational function.
Removable discontinuity (hole in graph)
Rational function graphs always have discontinuities (missing x-values) at the zeros of g(x).
Sometimes these discontinuities are removable discontinuities, meaning that it’s possible to
simplify gf (( xx)) so that the discontinuity no longer exists. In such a case, the graph of gf (( xx)) appears
to have a hole (a single missing point, illustrated with an open circle) at that discontinuity.
Limit concept and notation
If the graph of a function h(x) would pass through the point (a, b) except that there
is a hole at (a, b), then it’s not possible to write “h(a) = b” (because in fact, h(a) is
undefined). However, we can say that b is the limit of function h(x) near x = a.
The notation for this is: lim h( x) = b .
x →a
Example
Problem: Graph h( x) =
x 3 + x 2 − 3x − 3
. Write limit statements describing any discontinuities.
x +1
x 3 + x 2 − 3x − 3
= x 2 − 3 , except at x = –1 where h(x) is undefined.
x +1
So, the graph of h(x) looks like the graph of x2 – 3 except for a hole at x = –1.
Solution: Dividing gives
This limit statement describes the hole at (–1, –2):
lim h( x) = −2 .
x → −1
Name
November 12(G), or 18(H), 2014
Math 4 notes and problems
page 2
Exercises
Directions: For each of the following rational functions h(x):
•
Identify the domain of h(x). (Usually it will be all real numbers except for one or two that
you specify.)
•
Sketch a graph of h(x). (Do this by dividing or reducing to get a polynomial, graphing the
polynomial, then putting in any holes that h(x) should have.)
•
Write limit statements describing any discontinuities.
Checking your answers: Use your calculator to graph each function, but the calculator will
probably not accurately show the holes in the graph. Holes can often be located by pressing
[2nd][GRAPH] and looking at the table, which will show ERROR in the y column at a hole.
1. h(x) =
x 2 + 8 x + 15
x+5
2. h(x) =
x 3 − 6 x 2 + 8x
x−4
x 2 − 6x + 9
3. h(x) =
x−3
4. h(x) =
x5 − x3
x2 −1
5. h(x) =
x5 + x3
x2 +1
6. h(x) =
x 3 − 6 x 2 + 11x − 6
x 2 − 3x + 2
x 3 − 5 x 2 + 11x − 15
7. h(x) =
x 2 − 2x + 5
8. h(x) =
2 x 2 − 6 x − 20
x 2 − 3x − 10
9. h(x) =
x( x + 1)( x + 3)
x( x + 3)
10. h(x) =
( x − 2) 2 ( x − 3) 2 ( x − 4) 2
( x − 2) 2 ( x − 3)