Rational Functions 📘

Welcome, students! In this lesson, you will explore rational functions, a key part of the Functions topic in IB Mathematics: Analysis and Approaches HL. Rational functions appear in many real situations, such as speed, concentration, cost per item, and rates of change. They are especially useful because they combine ideas from polynomials and division, which creates interesting behavior like asymptotes, holes, and restricted domains.

Learning goals

By the end of this lesson, you should be able to:

• explain what a rational function is and use the correct terminology;
• identify the domain, intercepts, and asymptotes of a rational function;
• analyze graphs and connect features to algebraic structure;
• solve equations and inequalities involving rational functions;
• understand how rational functions connect to transformations, inverse functions, and broader function language.

Rational functions are a strong example of how algebra and graphs work together. A small change in the formula can create a big change in the graph, so careful reasoning matters. Let’s get started! 🚀

What is a rational function?

A rational function is a function that can be written as a ratio of two polynomials:

$$f(x)=\frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomials and $q(x)\neq 0$.

This definition is important because it tells us two things right away:

1. The function is built from polynomial expressions.
2. The denominator cannot be zero, so the domain may be restricted.

For example,

$$f(x)=\frac{2x+3}{x-1}$$

is a rational function because both $2x+3$ and $x-1$ are polynomials. However, $f(x)$ is undefined when $x=1$ because the denominator becomes $0$.

Another example is

$$g(x)=\frac{x^2-4}{x^2+x-6}$$

Since both numerator and denominator are polynomials, $g(x)$ is also rational.

Rational functions are different from polynomial functions because polynomials are defined for every real number, while rational functions may have excluded values. This difference is one reason rational functions are important in IB Mathematics: Analysis and Approaches HL.

Domain, intercepts, and simplifying carefully

The domain of a rational function is the set of all $x$-values for which the function is defined. For rational functions, the first step is usually to find values that make the denominator equal to $0$.

Example:

$$f(x)=\frac{x+1}{x-5}$$

The denominator is $x-5$, so $x\neq 5$. Therefore, the domain is all real numbers except $5$.

Now consider

$$h(x)=\frac{(x-2)(x+4)}{(x-2)(x+1)}$$

Algebraically, this simplifies to

$$h(x)=\frac{x+4}{x+1}$$

but only after noting that the original function is undefined at $x=2$ and $x=-1$.

This is where careful reasoning is essential, students. Even though $(x-2)$ cancels, the original function still has a restriction at $x=2$. That means the graph has a hole at $x=2$, not a removable point that simply disappears from reality.

Intercepts

The $x$-intercepts occur where $f(x)=0$. For a rational function, that happens when the numerator is $0$ and the denominator is not $0$.

For

$$f(x)=\frac{x+1}{x-5}$$

the numerator is $0$ when $x=-1$, so the $x$-intercept is $(-1,0)$.

The $y$-intercept occurs when $x=0$, provided the function is defined there.

For the same function,

$$f(0)=\frac{0+1}{0-5}=-\frac{1}{5}$$

so the $y$-intercept is $\left(0,-\frac{1}{5}\right)$.

If a numerator and denominator share a factor, check whether the factor creates a zero in the original function or a hole after simplification. This distinction is a major part of function analysis.

Asymptotes and end behavior

Rational functions often have vertical asymptotes and sometimes horizontal or oblique asymptotes.

Vertical asymptotes

A vertical asymptote occurs where the function grows without bound near a value of $x$ because the denominator approaches $0$ while the numerator does not also approach $0$ in the same way.

Example:

$$f(x)=\frac{2x+3}{x-1}$$

The denominator is $0$ at $x=1$, and the numerator is not $0$ there, so $x=1$ is a vertical asymptote.

Vertical asymptotes show where the graph may shoot upward or downward very quickly. In real life, this can model situations where a process becomes unstable, such as a cost per unit increasing sharply as production approaches a limiting value.

Horizontal asymptotes

A horizontal asymptote describes the long-term behavior as $x\to\infty$ or $x\to-\infty$.

For rational functions, compare the degrees of the numerator and denominator:

• if the degree of the numerator is less than the degree of the denominator, then $y=0$ is the horizontal asymptote;
• if the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients;
• if the degree of the numerator is greater by $1$, there may be an oblique asymptote.

Example:

$$f(x)=\frac{3x^2-1}{2x^2+5}$$

The degrees are both $2$, so the horizontal asymptote is

$$y=\frac{3}{2}$$

because the leading coefficients are $3$ and $2$.

Oblique asymptotes

If the numerator has degree exactly one more than the denominator, polynomial division may reveal a slanted asymptote.

Example:

$$f(x)=\frac{x^2+1}{x-2}$$

Dividing gives a linear quotient, and that quotient is the oblique asymptote.

These asymptotes are not just graph features; they summarize how a function behaves for very large $|x|$ values. This is a central function idea in IB: understanding behavior from structure.

Graphing rational functions using algebra and features

To sketch a rational function, use a systematic approach:

1. Find the domain.
2. Find the $x$- and $y$-intercepts.
3. Identify vertical asymptotes.
4. Identify horizontal or oblique asymptotes.
5. Check for holes.
6. Use test points to determine the sign and overall shape.

Let’s look at

$$f(x)=\frac{x+1}{x-2}$$

• Domain: $x\neq 2$
• $x$-intercept: $x=-1$, so $(-1,0)$
• $y$-intercept: $f(0)=\frac{1}{-2}=-\frac{1}{2}$, so $\left(0,-\frac{1}{2}\right)$
• Vertical asymptote: $x=2$
• Horizontal asymptote: $y=1$ because the degrees are equal and the leading coefficients are both $1$

This graph has two branches, one on each side of $x=2$. As $x$ gets very large, the graph approaches $y=1$. As $x$ gets close to $2$, the function values increase or decrease very rapidly.

Now consider a function with a hole:

$$g(x)=\frac{(x-3)(x+2)}{(x-3)(x-1)}$$

After simplification,

$$g(x)=\frac{x+2}{x-1}$$

but the original function is undefined at $x=3$ and $x=1$. So the graph has a vertical asymptote at $x=1$ and a hole at $x=3$.

This is a classic IB-style reasoning task: the simplified expression helps you analyze the graph, but the original expression tells you the true domain.

Equations and inequalities involving rational functions

Rational functions often appear in equations and inequalities. A reliable method is to rewrite everything on one side and use algebraic reasoning carefully.

Solving equations

Example:

$$\frac{x+1}{x-2}=3$$

Multiply both sides by $x-2$, remembering that $x\neq 2$:

$$x+1=3(x-2)$$

$$x+1=3x-6$$

$$7=2x$$

$$x=\frac{7}{2}$$

Since $\frac{7}{2}\neq 2$, this solution is valid.

Solving inequalities

Example:

$$\frac{x-1}{x+2}>0$$

First, find critical values: $x=1$ makes the numerator $0$, and $x=-2$ makes the denominator $0$.

These values split the number line into intervals:

• $(-\infty,-2)$
• $(-2,1)$
• $(1,\infty)$

Test each interval:

• If $x=-3$, then $\frac{-4}{-1}>0$
• If $x=0$, then $\frac{-1}{2}<0$
• If $x=2$, then $\frac{1}{4}>0$

So the solution is

$$(-\infty,-2)\cup(1,\infty)$$

Notice that $x=-2$ is excluded because the function is undefined there, and $x=1$ is excluded because the inequality is strict $>0$.

This kind of interval testing is very important in HL work because sign changes matter.

Connections to transformations and inverses

Rational functions also connect to transformations. Starting from

$$f(x)=\frac{1}{x}$$

many rational graphs can be created by transformations such as shifts, stretches, and reflections.

For example,

$$g(x)=\frac{1}{x-3}+2$$

is the graph of $\frac{1}{x}$ shifted right $3$ units and up $2$ units.

• vertical asymptote: $x=3$
• horizontal asymptote: $y=2$

This transformation view is powerful because it helps you recognize graphs quickly.

Rational functions can also connect to inverse functions. If a rational function is one-to-one on a restricted domain, it may have an inverse. For example, restricting

$$f(x)=\frac{1}{x}$$

to positive $x$-values gives a one-to-one function whose inverse is itself:

$$f^{-1}(x)=\frac{1}{x}$$

Inverse and composite function ideas help connect rational functions to the wider Functions topic. They show that functions are not isolated formulas; they are objects with structure, behavior, and relationships.

Conclusion

Rational functions are a major part of the IB Mathematics: Analysis and Approaches HL Functions topic because they combine algebraic manipulation with graph interpretation. students, you have seen how to identify domains, intercepts, holes, asymptotes, and long-term behavior. You have also seen how to solve equations and inequalities involving rational expressions and how transformations create families of rational graphs.

The main lesson is this: the formula of a rational function tells you much more than a rule for calculation. It reveals where the function exists, how it behaves, and what its graph looks like. That is exactly the kind of connected reasoning IB mathematics values. ✨

Study Notes

• A rational function has the form $f(x)=\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x)\neq 0$.
• The domain excludes values that make the denominator $0$.
• An $x$-intercept occurs when the numerator is $0$ and the denominator is not $0$.
• A $y$-intercept is found by evaluating the function at $x=0$ if possible.
• A vertical asymptote usually occurs where the denominator is $0$ after simplification, and the numerator is not $0$ there.
• A canceled factor can create a hole, not an asymptote.
• If the numerator degree is less than the denominator degree, the horizontal asymptote is $y=0$.
• If the degrees are equal, the horizontal asymptote is the ratio of leading coefficients.
• If the numerator degree is one more than the denominator degree, there may be an oblique asymptote.
• To solve rational inequalities, use critical values and test intervals.
• Rational functions connect to transformations of $f(x)=\frac{1}{x}$ and to inverse/composite function ideas.
• Careful use of the original expression matters, even after simplification.