Answer:
A rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials, nor the values taken by the function, are necessarily rational numbers.
Any function of one variable,
x
, is called a rational function if, and only if, it can be written in the form:
f
(
x
)
=
P
(
x
)
Q
(
x
)
where
P
and
Q
are polynomial functions of
x
and
Q
(
x
)
≠
0
.
Note that every polynomial function is a rational function with
Q
(
x
)
=
1
. A function that cannot be written in the form of a polynomial, such as
f
(
x
)
=
sin
(
x
)
, is not a rational function. However, the adjective “irrational” is not generally used for functions.
A constant function such as
f
(
x
)
=
π
is a rational function since constants are polynomials. Note that the function itself is rational, even though the value of
f
(
x
)
is irrational for all
x
.
The Domain of a Rational Function
The domain of a rational function
f
(
x
)
=
P
(
x
)
Q
(
x
)
is the set of all values of
x
for which the denominator
Q
(
x
)
is not zero.
For a simple example, consider the rational function
y
=
1
x
. The domain is comprised of all values of
x
≠
0
.
Domain restrictions can be calculated by finding singularities, which are the
x
-values for which the denominator
Q
(
x
)
is zero. The rational function is not defined for such
x
-values, and these values are excluded from the domain set of the function.
Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational functions. Singularity occurs when the denominator of a rational function equals
0
, whether or not the linear factor in the denominator cancels out with a linear factor in the numerator.
Example 1
Consider the rational function
f
(
x
)
=
(
x
2
−
3
x
−
2
)
(
x
2
−
4
)
The domain of this function includes all values of
x
, except where
x
2
−
4
=
0
.
We can factor the denominator to find the singularities of the function:
x
2
−
4
=
(
x
+
2
)
(
x
−
2
)
Setting each linear factor equal to zero, we have
x
+
2
=
0
and
x
−
2
=
0
. Solving each of these yields solutions
x
=
−
2
and
x
=
2
; thus, the domain includes all
x
not equal to
2
or
−
2
. This can be seen in the graph below.
The function is always increasing and has vertical asymptotes at x=-2 and x=2. From x=negative infinity to x=-2, it increases from the positive horizontal asymptote to infinity. From x=-2 to x=2, it increases from negative infinity to infinity. From x=2 to x=infinity, it increases from negative infinity to the positive horizontal asymptote.
The domain of a function: Graph of a rational function with equation
(
x
2
–
3
x
−
2
)
(
x
2
–
4
)
. The domain of this function is all values of
x
except
+
2
or
−
2
.
Note that there are vertical asymptotes at
x
-values of
2
and
−
2
. This means that, although the function approaches these points, it is not defined at them.
Example 2
Consider the rational function
f
(
x
)
=
(
x
+
3
)
(
x
2
+
2
)
The domain of this function is all values of
x
except those where
x
2
+
2
=
0
. However, for
x
2
+
2
=
0
,
x
2
would need to equal
−
2
. Since this condition cannot be satisfied by a real number, the domain of the function is all real numbers.
Asymptotes
A rational function can have at most one horizontal or oblique asymptote, and many possible vertical asymptotes; these can be calculated.
LEARNING OBJECTIVES
Determine when the asymptote of a rational function will be horizontal, oblique, or vertical
KEY TAKEAWAYS
Key Points
An asymptote of a curve is a line, such that the distance between the curve and the line approaches zero as they tend to infinity.
There are three kinds of asymptotes: horizontal, vertical and oblique.
A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.
Vertical asymptotes occur at singularities of a rational function, or points at which the function is not defined. They only occur at singularities where the associated linear factor in the denominator remains after cancellation.
The existence of a horizontal or oblique asymptote depends on the degrees of polynomials in
the numerator and denominator.
Key Terms
asymptote: A straight line which a curve approaches arbitrarily closely, as it goes to infinity.
oblique: Not erect or perpendicular; neither parallel to, nor at right angles from, the base; slanting; inclined.
rational function: Any function whose value be expressed as the quotient of two polynomials (where the polynomial in the denominator is not zero).
Step-by-step explanation:
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