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Rational functions
Reciprocal functions, quadratic denominators, graphs of rational linear functions
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Exercises
No exercises available for this concept.
Reciprocal functions, quadratic denominators, graphs of rational linear functions
Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)
No exercises available for this concept.
The reciprocal function is defined by f(x)=x1​.
Notice that f(x) is not defined for x=0. In fact, since x1​ gets very large as x approaches 0, f(x) has a vertical asymptote at x=0.
And since for very large x, x1​ approaches zero, there is also a horizontal asymptote y=0.
Notice also that x1​1​=x, so f(x)=x1​ is self-inverse.
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A linear rational function has the form
When the denominator is zero the graph will have a vertical asymptote:
And as x gets very large, the +b and +d can be ignored:
So there is a horizontal asymptote at y=ca​.
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When a rational function is of the form
there is a vertical asymptote at
By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation
for some constant C determined during the polynomial division.
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When a rational function is of the form
There will be vertical asymptotes when the quadratic cx2+dx+e=0.
The horizontal asymptote will simply be y=0 since cx2 dominates ax when x is very large.
Additionally, there will be an x-intercept at x=−ab​, when the numerator changes sign.
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Note that if the numerator and denominator share a root (ie x=−ab​ is a root of the denominator), then there will only be one asymptote and a "hole" on the x-axis. This has never shown up on exams.