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Rational functions
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Reciprocal functions, quadratic denominators, graphs of rational linear functions
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The reciprocal function 1/x
SL AA 2.8
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 x11=x, so f(x)=x1 is self-inverse.
Graphs of linear rational functions
SL AA 2.8
A linear rational function has the form
f(x)=cx+dax+b
When the denominator is zero the graph will have a vertical asymptote:
cx+d=0⇒x=−cd🚫
And as x gets very large, the +b and +d can be ignored:
y=f(x)≈cxax=ca🚫
So there is a horizontal asymptote at y=ca.
Rationals with oblique asymptotes
AHL 2.13
When a rational function is of the form
f(x)=dx+eax2+bx+c
there is a vertical asymptote at
x=−de🚫
By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation
y=dax+C
for some constant C determined during the polynomial division.
Rationals with quadratic denominator
AHL 2.13
When a rational function is of the form
f(x)=cx2+dx+eax+b
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.
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.
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