Reciprocal Functions
Lesson · Beginner
Algebra: Functions
A reciprocal function has the basic form
Equivalently,
So as the magnitude of x increases, the magnitude of y decreases while their product remains constant.
For example,
contains the points
because in each case
For
we cannot divide by zero, so
Also, because a ≠ 0,
Therefore,
For the basic reciprocal function
the vertical asymptote is
and the horizontal asymptote is
As x gets closer to 0, the magnitude of y becomes larger.
As the magnitude of x becomes larger, y gets closer to 0.
Find the asymptotes, center, domain, and range of
The vertical asymptote is
The horizontal asymptote is
Therefore, the center is
The domain is
and the range is
Find all values of c such that
is tangent to
Set the expressions equal:
Multiply by x:
For tangency,
so
Thus,
The line
intersects
at points A and B. If the midpoint of AB lies on
find c.
At the intersections,
Multiplying by x,
so
By Vieta's formulas,
Since both points lie on y = -x + c,
Therefore the midpoint is
The midpoint lies on
Thus
so
Hence
and therefore
The equation
can be rewritten as
Therefore, its asymptotes are immediately