asymptotes. rational function a rational function is one that can be written in the form where p(x)...
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Asymptotes
Rational function
• A rational function is one that can be written in the form
where p(x) and q(x) are polynomials and q(x) is not the zero polynomial. In general, the domain of a rational function includes all real numbers except x-values which make the denominator zero.
( )( )
( )
p xf x
q x
Find domain
• Find the domain of each of the following:
1f(x)x 2
x-1f(x)
(x 1)(x 2)
xf(x)2x 1
Answer• Set denominator not equal to 0
2
2 02
xx
1f(x)x 2
1, 2
01; 2x x
x -1f(x)(x 1)(x 2)
(x 1)(x 2)0
xf(x)2x 1
2x 1
2x 1
Vertical Asymptotes• At the value(s) for which the domain is undefined, there will
be one or more vertical asymptotes. List the vertical asymptotes for the following.
1f(x)x 2
x-1f(x)
(x 1)(x 2)
xf(x)2x 1
Answer• Set denominator = 0
2 02
xx
1f(x)x 2
01, 2x
x -1f(x)(x 1)(x 2)
(x 1)(x 2)0
xf(x)2x 1
2x 1
2x 1
none
Graphically
• The figure below shows the graph of The equation of the vertical asymptote is x=-2.
. 1f(x)x 2
Summary• The graph of a function has vertical asymptote(s) at the
discontinuity of the function. (set the denominator = 0)
• If the degree of the numerator is = to the degree of the denominator, the function has a horizontal asymptote. ( y = ratio of lead coefficients)
• • If the degree of the numerator is < the degree of the
denominator, the function has a horizontal asymptote. (y = 0, the x-axis).
• • If the degree to the numerator is > the degree of the
denominator, the function does not have a horizontal asymptote.
Find asymptotes• Name the vertical and horizontal asymptotes for
1 2x+1f(x)x
Answer• Domain:
• Degree of numerator: 1
• Degree of denominator: 1
• HA: y = 2
• VA: x = -1
, 1x
Find asymptotes• Name the vertical and horizontal asymptotes for
2
2 85x +1f(x)2x
Answer• Domain:
• Degree of numerator: 2
• Degree of denominator: 2
• HA: y = 5/2
• VA: x = -2,2
, 2, 2x 2
2
2(
2 8 0
2( 4) 0
2)( 2) 0
2,2
x
x
x
x
x
Find asymptotes• Name the vertical and horizontal asymptotes for
2
41x
f(x)
Answer• Domain:
• Degree of numerator: 0
• Degree of denominator: 2
• HA: y = 0
• VA: none
Find asymptotes• Name the vertical and horizontal asymptotes for
2
19
xx
f(x)
Answer• Domain:
• Degree of numerator: 1
• Degree of denominator: 2
• HA: y = 0
• VA: x=3,-3
3, 3 2
( 3)( 3)
3 3
0
,
9
0x x
x
x
Find asymptotes• Name the vertical and horizontal asymptotes for
2 92
xx
f(x)
Answer• Domain:
• Degree of numerator: 2
• Degree of denominator: 1
• HA: none
• VA: x=-2
2
Find asymptotes• Name the vertical and horizontal asymptotes for
3 23 5 4 53 1
x x xx
f(x)
Answer• Domain:
• Degree of numerator: 3
• Degree of denominator: 1
• HA: none
• VA: x=-1/3
13 3 1 0
3 1
1 / 3
x
x
x