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**Unformatted text preview: **. Putting all of our work together yields the graph below.
y
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8
7
6
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4
3
2
1 (−)
−2 13 (+) (−) 0 (+)
−1 1
−2 Also called an ‘oblique’ asymptote in some texts. −4 −3 −1 −1
−2
−3
−4
−5
−6
−7
−8
−9
−10
−11
−12
−13
−14 1 2 3 4 x 4.2 Graphs of Rational Functions 257 We could ask whether the graph of y = h(x) crosses its slant asymptote. From the formula
3
3
h(x) = 2x − 1 + x+2 , x = −1, we see that if h(x) = 2x − 1, we would have x+2 = 0. Since this will
never happen, we conclude the graph never crosses its slant asymptote.14
We end this section with an example that shows it’s not all pathological weirdness when it comes
to rational functions and technology still has a role to play in studying their graphs at this level.
Example 4.2.4. Sketch the graph of r(x) = x4 + 1
.
x2 + 1 Solution.
1. The denominator x2 + 1 is never zero so the domain is (−∞, ∞).
2. With no real zeros in the denominator, x2 + 1 is an irreducible quadratic. Our only hope of
reducing r(x) is if x2 + 1 is a factor of x4 + 1. Performing long divisi...

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