Properties of Rational Functions in Precalculus
These are my notes on properties of rational functions in precalculus.
Ratios of integers are called rational numbers. Ratios of polynomial functions
are called rational functions. A rational function is of the form
\(r(x)=\frac{p(x)}{q(x)}\). P and Q are polynomial functions and q is not the
zero polynomial. The domain of a rational function is the set of all real
numbers except those for which the denominator is zero.
Example 1
Find the domain of a rational function
1. The domain of \(r(x)=\frac{2x^2-4}{x+5}\) is the set of all real numbers
except -5. The domain is \({x|x\not=-5}\).
2. The domain of \(r(x)=\frac{1}{x^2-4}\) is the set of all real numbers except
-2 so the domain is \({x|x\not=-2,x\not=2}\).
3. The domain of \(r(x)=\frac{x^3}{x^2+1}\) is the set of all real numbers.
4. The domain of \(r(x)=\frac{3}{x^2+2}\) is the set of all real numbers.
5. The domain of \(r(x)=\frac{x^2-1}{x-1}\) is the set of all real numbers
except 1 so the domain is \({x|x\not=1}\).
If \(r(x)=\frac{p(x)}{q(x)}\) is a rational function and if p and q have no
common factors, then the rational function is said to be in lowest terms. For a
rational function in lowest terms, the real zeros, if any, of the numerator are
the x-intercepts of the graph will play a major role in the graph. The real
zeros of the denominator also play a major role in the graph.
Example 2
Graphing \(y=\frac{1}{x^2}\)
Solution:
The domain is the set of all real numbers except 0. The graph has no y-intercept
because x can never equal 0. The graph has no x-intercept because the equation
\(y=0\) has no solution. Therefore, the graph will not cross either of the
coordinate axes. Because:
\(h(-x) = \frac{1}{(-x)^2} = \frac{1}{x^2} = h(x)\)
It is an even function, so its graph is symmetric with respect to the y-axis.
Example 3
Using transformations to graph a rational function
Graph the rational function \(r(x) = \frac{1}{(x-2)^2 + 1}\)
Solution:
First, we take note of the fact that the domain of r is the set of all real
numbers except x=2. To graph r, we start with the graph of \(y=\frac{1}{x^2}\).
Use a calculator to graph the function.
Asymptotes
In the previous graph, notice that as the values of x become more negative as x
approaches negative infinity, the values approach 1. In fact, we can conclude
the following:
1. As x approaches negative infinity, the values of r(x) approach 1
2. As x approaches 2, the values of r(x) approach infinity
3. As x approaches infinity, the values of r(x) approach 1
This behavior of the graph is shown by the vertical line x=2 and the horizontal
line y=1. These lines are called asymptotes of the graph.
A horizontal asymptote, when it occurs, describes the end behavior of the graph
as x approaches infinity or as x approaches negative infinity. The graph of a
function may intersect a horizontal asymptote. A vertical asymptote, when it
occurs, describes the behavior of the graph when x is close to some number. The
graph of a function will never intersect a vertical asymptote.
There is a third possibility. If the value of a rational function approaches a
linear expression, then the line y=ax+b is an oblique asymptote. An oblique
asymptote, when it occurs, describes the end behavior of the graph. The graph of
a function may intersect an oblique asymptote.
Find the Vertical Asymptotes of a Rational Function
The vertical asymptotes of a rational function, in lowest terms, are located at
the real zeros of the denominator. Suppose that r is a real zero of q, so x-r is
a factor of q. As x approaches r, the values of x-r approach 0, causing the
ratio to become unbounded, or head towards infinity. Based on the definition, we
conclude that the line x=r is a vertical asymptote.
Example 4
Finding vertical asymptotes
Find the vertical asymptotes of the graph of each rational function
1. \(r(x)=\frac{x}{x^2-4}\)
2. \(r(x)=\frac{x+3}{x-1}\)
3. \(r(x)=\frac{x^2}{x^2+1}\)
4. \(r(x)=\frac{x^2-9}{x^2+4x-21}\)
Solution:
1. The function is in lowest terms and the zeros of the denominator are -2 and
The lines x=-2 and x=2 are the vertical asymptotes of the graph.
2. The function is in lowest terms and the only zero of the denominator is 1.
The line x=1 is the vertical asymptote of the graph.
3. The function is in lowest terms and the denominator has no real zeros,
because the equation has no real solutions. The graph has no vertical asymptotes
4. Factor to get it to lowest terms.
\(r(x)=\frac{x^2-9}{x^2+4x-21} = \frac{(x+3)(x-3)}{(x+7)(x-3)} =
\frac{x+3}{x+7}\) The only zero of the denominator in lowest terms is -7. The
line x=-7 is the only vertical asymptote of the graph.
Rational functions can have no vertical asymptotes, one vertical asymptote, or
multiple vertical asymptotes. However, the graph of a rational function will
never intersect any of its vertical asymptotes.
Find the Horizontal or Oblique Asymptotes of a Rational Function
The procedure for finding horizontal and oblique asymptotes is somewhat more
involved. To find such asymptotes, we need to know how the values of a function
behave as x a[[roaches negative infinity or as x approaches infinity.
If a rational function is proper, that is, if the degree of the numerator is
less than the degree of the denominator, then as x approaches negative infinity
or as x approaches infinity the value of the function approaches 0.
Consequently, the line y=0(x-axis) is a horizontal asymptote of the graph. If a
rational function is proper, the line y=0 is a horizontal asymptote of its graph.
Example 5
Finding Horizontal Asymptotes
Find the horizontal asymptotes, if any, of the graph of:
\(f(x)=\frac{x-12}{4x^2+x+1}\)
Solution:
Since the degree of the numerator is 1, it is less than the degree of the
denominator which is 2. So this is a proper function. That means that the line
y=0 is a horizontal asymptote of the graph.
To see why y=0 is a horizontal asymptote of the function, we need to investigate
the behavior as x approaches negative infinity and as x approaches infinity.
When |x| is unbounded, the numerator, which is x-12, can be approximated by the
power function y=x, while the denominator, which is \(4x^2+x+1\), can be
approximated by the power function \(y=4x^2\). We find:
\(f(x)=\frac{x-12}{4x^2+x+1} = \frac{x}{4x^2} = \frac{1}{4x}\) as it approaches
0. This shows that the line y=0 is a horizontal asymptote of the graph.
If a rational function is improper, or if the degree of the numerator is greater
than the degree of the denominator, we must use long division to write the
rational function as the sum of a polynomial plus a proper rational function.
1. If f(x)=b, a constant, the line y=b is a horizontal asymptote of the graph
2. If f(x)=ax+b, the line y=ax+b is an oblique asymptote of the graph
3. In all other cases, the graph of r approaches the graph of f, and there are
no horizontal or oblique asymptotes.
Example 6
Finding Horizontal or Oblique Asymptotes
Find the horizontal or oblique asymptote of the graph of:
\(f(x)=\frac{3x^4-x^2}{x^3-x^2+1}\)
Solution:
Since the degree of the numerator, 4, is larger than the degree of the
denominator, 3, the rational function is improper. To find any horizontal or or
oblique asymptotes, use long division to get 3x+3, which is the oblique
asymptote.
Example 7
Finding Horizontal or Oblique Asymptotes
Find asymptotes of:
\(f(x)=\frac{8x^2-x+2}{4x^2-1}\)
Solution:
Since the degree of the numerator, 2, equals the degree of the denominator,2,
the rational function is improper. To find any horizontal or oblique asymptotes,
use long division. We get y=2 as a horizontal asymptote of the graph.
We note that the quotient 2 obtained by long division is the quotient of the
leading coefficients of the numerator polynomial and the denominator polynomial.
This means that we can avoid the long division process for rational functions
whose numerator and denominator are of the same degree and conclude that the
quotient of the leading coefficients will give us the horizontal asymptote.
Example 8
Finding the Horizontal or Oblique Asymptotes
\(f(x)=\frac{2x^5-x+2}{x^3-1}\)
Solution:
Since the degree of the denominator, 5, is larger than the degree of the
denominator, 3, the rational function is improper. To find any horizontal or
oblique asymptotes, use long division. \(2x^2-1\). Since this is not a linear
function, the function has no horizontal or oblique asymptotes.
1. If the degree of the numerator is less than the degree of the denominator, it
is a proper rational function, and the graph will have the horizontal asymptote
y=0.
2. If the degree of the numerator is greater than or equal to the degree of the
denominator, then it is an improper function. Use long division.
A. if the degree of the numerator equals the degree of the denominator, the
quotient obtained will be the number\(\frac{a_{n}}{b_{m}}\) and the line
\(y=\frac{a_{n}}{b_{m}}\) is a horizontal asymptote.
B. If the degree of the numerator is more than the degree of the
denominator, the quotient obtained is of the form ax+b and the line y=ax+b
is an oblique asymptote.
C. If the degree of the numerator is two or more than the degree of the
denominator, the quotient obtained is a polynomial of degree 2 or higher,
and the function has neither a horizontal nor an oblique asymptote. In this
case, for |x| unbounded, the graph will behave like the graph of the
quotient.