Rational functions are quotients or fractions of polynomials. (Polynomials are rational functions, since they can be written in a fraction form with \(1\) as the denominator.) So, it is not surprising that analyzing rational function follows the same plan as for polynomials.

\(\blacktriangleright \) Domain: The natural or implied domain of a rational function is all real numbers, except those that make the denominator equal to \(0\).

Of course, any particular rational function may be defined with a stated subset of the real numbers as the domain, in which case we say that it is a restricted rational function.

Following our analysis of polynomials, we would like to factor, collect common factors, and reduce or simplify.

Here, the roots are distinct, which means that \(r_i \ne r_j\) when \(i \ne j\), and \(s_i \ne s_j\) when \(i \ne j\), and \(r_i \ne s_j\) for all \(i\) and \(j\).

Let \(R\) be defined as follows.

\[ R(x) = \frac {(x-3)(x-1)}{x-1} \]

Just thinking algebraically, we might simplify this to \(R(x) = x-3\). However, thinking functionally, these two formulas have different natural domains.

We have to remember that that the domain of \(R\) is still \(\left ( -\infty , \answer {1} \right ) \cup \left ( \answer {1},\infty \right )\) no mater what algebraic modifications have been made.

\(R(x)\) and \(p(x) = x-3\) are different functions, since they have different natural domains. \(R(x) = x - 3\), provided we note that \(x \ne 1\).

The graphs of \(R(x)\) and \(p(x) = x-3\) are the same, except at \(x=1\). The graph of \(p(x) = x-3\) includes the point \((1, -2)\), while the graph of \(R(x)\) omits this point.

The graph of \(p(x) = x-3\) is a line, while the graph is the same line with a hole in it at \(\left ( 1, \answer {-2} \right )\).

Graph of \(y = R(x)\).

\(1\) does not cause a problem for \(x-3\), but that isn’t the formula for \(R\).

\(R(x) = \frac {(x-3)(x-1)}{(x-1)}\) is equal to \(x-3\), provided \(x \ne 1\). We cannot reinsert \(1\) after simplifying.

We prefer working with a simplified formula. We just need to keep track of how simplifying affects factors.

Assuming a simplified formula...

Note: A simplified formula assures us that the zeros of the numerator polynomial are all different from the zeros of the denominator polynomial. (Otherwise, we would have removed the common factors.)

\(\blacktriangleright \) Roots and Zeros:
Zeros or Roots of a rational function are the zeros or roots of the numerator polynomial. A polynomial function behaves in one of two ways around a root.

• If the multiplicity is odd then the function changes sign over the root. The graph crosses over the horizontal axis at the corresponding intercept.
• If the multiplicity is even then the function does not change sign over the root. The graph does not cross over the horizontal axis at the corresponding intercept. Instead, it bounces back in the direction from which it came.

\(\blacktriangleright \) Singularities:
Singularities of a rational function are the zeros of the denominator polynomial. Vertical asymptotes usually represent these types of singularities graphically.

• If the multiplicity is odd then the function changes sign over the singularity. The graph moves to the other end of the corresponding vertical asymptote.
• If the multiplicity is even then the function does not change sign over the singularity. The graph does not move to the other end of the corresponding vertical asymptote.

\(\blacktriangleright \) Hole:
Another type of singularity for rational functions is a hole. If a common factor between the numerator and denominator have the same multiplicity, then they will be completely removed from the formula upon simplification. In this case, the rational function behaves nicely, except that particular number is removed from the domain. We have a hole in the graph. The new, simplified formula has a value for the missing factors. However, we cannot use it, because the factors were initially there, thus removing the zero from the domain. Therefore, we end up with a hole in the graph.

Note: Remember, the original formula dictates the domain. The simplified form might have a factor in the numerator, indicating a zero. However, if the original formula also had a matching denominator factor, then there will be a hole in the graph rather than an intercept.

Note: Removeable singularities show up as holes in the graph. There is no domain number corresponding to the point that has been removed. The hole can be filled by inserting a single point. The singularity can be removed by defining the function at a single number. This would “remove” the singularity.

Of course “fixing” the hole of singularity creates a new and different function. This new and different function is exactly the same as the original function, except for one single ordered pair.

Note: Removeable discontintinuities are just like removeable singlularities, except there is a point on the graph. It is just somewhere else besides filling in the hole. The hole or discontinuity can be “removed” by re-defining the function at that one domain number to fill in the hole.

Of course moving the dot and “filling in” the hole of discontinuity creates a new and different function. This new and different function is exactly the same as the original function, except for one single pair.

We must always remember the original formula when it comes to the domain. Simplification cannot change the domain.

\(\blacktriangleright \) Continuity:
For the most part, rational functions are relatively nice functions. They are continuous everywhere in their domain. They have no discontinuities. The singularities correspond to vertical asymptotes or removeable singularities, both of which we understand.

Of course, we are always on the lookout for common factors, which are represented with holes in the graph.

\(\blacktriangleright \) End-Behavior:
In addition to identifying zeros, roots, dicontinuities, singularities, maximums, and minimums, we would also like to describe any pattern the function takes on as the domain heads to infinity or negative infinity. These patterns we call the “end-behavior” of the function.

In the case of a rational function there are three types of end-behavior, which can be ascertained by comparing the leading terms of the polynomials in the numerator and denominator.

• If the degrees of the leading terms are equal then the end-behavior is the quotient of the leading coefficients.

  \[ \lim \limits _{x \to \infty } R(x) = \frac {a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0}{b_n x^n + b_{n-1} x^{n-1} + \cdots + b_2 x^2 + b_1 x + b_0} = \frac {a_n}{b_n} \]
  \[ \lim \limits _{x \to -\infty } R(x) = \frac {a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0}{b_n x^n + b_{n-1} x^{n-1} + \cdots + b_2 x^2 + b_1 x + b_0} = \frac {a_n}{b_n} \]

• If the degree of numerator polynomial is greater than the degree of the denominator polynomial, then the rational function becomes unbounded.

  If \(n > m\), then

  \[ \lim \limits _{x \to \infty } R(x) = \frac {a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \cdots + b_2 x^2 + b_1 x + b_0} = \pm \infty \]
  \[ \lim \limits _{x \to -\infty } R(x) = \frac {a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \cdots + b_2 x^2 + b_1 x + b_0} = \pm \infty \]

  The sign depends on the signs of the leading coefficients and the degrees of the polynomials.

• If the degree of denominator polynomial is greater than the degree of the numerator polynomial, then the rational function tends to \(0\).

  If \(n < m\), then

  \[ \lim \limits _{x \to \infty } R(x) = \frac {a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \cdots + b_2 x^2 + b_1 x + b_0} = 0 \]
  \[ \lim \limits _{x \to -\infty } R(x) = \frac {a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \cdots + b_2 x^2 + b_1 x + b_0} = 0 \]

\(\blacktriangleright \) Graph:
Graphs of rational functions are nice. They are smooth. They do not have corners or spikes or jump breaks. They do have vertical asymptotes, which we understand.

Once we have the roots of the numerator and denominator polynomials, then we can plot the intercepts and asymptotes. Then we can smoothly connect everything according to their multiplicities and have a pretty good sketch of the shape of the graph.

With a basic general shape, we can estimate critical numbers and identify the type of extrema values.

Of course, we are always on the lookout for common factors, which are represented with holes in the graph.

\(\blacktriangleright \) Extrema:
If we have a derivative, then we can attempt to locate exact values of critical numbers. Without the derivative, we turn to technology for some assistance in approximating critical numbers.

\(\blacktriangleright \) Rate of Change:
The critical numbers and singularities partition the real line into intervals where the rational function increases or decreases. We can then list these intervals.