# Reciprocal and rational functions

> IB Mathematics AA SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u2-reciprocal-and-rational-functions/

This module covers reciprocal functions and simple linear-over-linear rational functions for IB AA SL. You will learn to find key features, sketch graphs, and solve rational equations, aligned to exam expectations.

**Prerequisites:** [Polynomial functions and domain/range concepts](https://www.owlsprep.com/study/ib-math-aa-sl-u2-polynomial-functions/); Solving linear and quadratic equations

## Learning objectives

- Identify and define reciprocal and simple rational functions
- Find domain, range, and equations of asymptotes
- Calculate intercepts and sketch graphs of these functions
- Solve rational equations and identify extraneous solutions

## Reciprocal Functions: Key Features

**Reciprocal Function** — A transformed reciprocal function, formed by translating and stretching the basic reciprocal function $y = \frac{1}{x}$. The denominator cannot equal zero, so there is a restriction on the domain.

*Notation:* $f(x) = \frac{k}{x-h} + v, k \neq 0$

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

For any reciprocal function in the standard form above: the vertical asymptote (where the function is undefined) occurs at $x=h$, and the horizontal asymptote, approached as $x$ grows very large in magnitude, occurs at $y=v$.

**Worked example:** For $f(x) = \frac{4}{x+2} - 1$, find: (i) domain, (ii) asymptote equations, (iii) range.

1. Find the excluded domain value by setting the denominator to zero:
2. $$x+2 = 0 \implies x = -2$$
3. Domain is all real numbers except $x=-2$, written as: $x \in \mathbb{R}, x \neq -2$
4. Match to standard form $\frac{k}{x-h} + v$: here $h=-2$, $v=-1$. So:
5. Vertical asymptote: $x = -2$, Horizontal asymptote: $y=-1$
6. The function can never equal the horizontal asymptote value, so range is: $y \in \mathbb{R}, y \neq -1$

> **tip**
>
> Always shift your graph according to $(h,v)$: the center of the two branches is at the point $(h,v)$ where the asymptotes cross.

> **Exam tip:** Always write asymptotes as full equations, not just numbers. Examiners will not award marks for just writing $-2$ instead of $x=-2$.

*Calculator:* forbidden

## Linear-over-Linear Rational Functions

**Simple Rational Function** — A rational function where both numerator and denominator are linear polynomials, the most common form tested in IB AA SL exams.

*Notation:* $f(x) = \frac{ax+b}{cx+d}, c \neq 0$

To find asymptotes for this form: 
1. Vertical asymptote: Set denominator equal to zero, solve for $x$. 
2. Horizontal asymptote: Divide numerator and denominator by $x$, the limit as $x \to \pm \infty$ is $\frac{a}{c}$, so the asymptote is $y = \frac{a}{c}$.

**Worked example:** For $f(x) = \frac{3x + 6}{x - 2}$, find all asymptotes and the x and y intercepts.

1. Find vertical asymptote: set denominator to zero:
2. $$x - 2 = 0 \implies x = 2$$
3. Vertical asymptote: $x=2$
4. Find horizontal asymptote: leading coefficient of numerator is 3, leading coefficient of denominator is 1:
5. $$y = \frac{3}{1} = 3$$
6. Horizontal asymptote: $y=3$
7. Find y-intercept: substitute $x=0$:
8. $$f(0) = \frac{0 + 6}{0 - 2} = -3 \implies \text{y-intercept } (0, -3)$$
9. Find x-intercept: set numerator equal to zero:
10. $$3x + 6 = 0 \implies x = -2 \implies \text{x-intercept } (-2, 0)$$

*Calculator:* forbidden

## Solving Rational Equations

To solve a rational equation, first eliminate denominators by multiplying both sides by the lowest common denominator, then solve the resulting linear or quadratic equation. Always check your solutions against the original domain restriction.

> **warning**
>
> Multiplying by a denominator can introduce extraneous solutions that make the original denominator zero, which are not valid. Exam markers always allocate a mark for checking this.

**Worked example:** Solve the equation $\frac{2x}{x-1} = 5 + \frac{2}{x-1}$

1. First note the domain restriction: denominator $x-1 \neq 0 \implies x \neq 1$
2. Multiply every term by $(x-1)$ to eliminate denominators:
3. $$2x = 5(x-1) + 2$$
4. Expand and simplify:
5. $$2x = 5x - 5 + 2 \implies -3x = -3 \implies x = 1$$
6. Check against domain restriction: $x=1$ is not allowed, so this is extraneous
7. Conclusion: the equation has no real solutions

> **Exam tip:** If a question asks for the number of solutions, always remember to count only valid non-extraneous solutions.

## Common pitfalls

- **Wrong:** Writing only the number for an asymptote, e.g. writing $2$ instead of $x=2$
  - Why it fails: Examiners require the full equation of the asymptote line, not just the intercept value
  - Correct: Always write vertical asymptotes as $x=a$ and horizontal asymptotes as $y=b$
- **Wrong:** Forgetting to check for extraneous solutions after solving a rational equation
  - Why it fails: Multiplying by the denominator can introduce invalid solutions that do not satisfy the original equation
  - Correct: Always check any solution against the original domain, discard any that make a denominator zero
- **Wrong:** Confusing vertical and horizontal asymptote formulas for $\frac{ax+b}{cx+d}$, writing $y=-\frac{d}{c}$
  - Why it fails: Mixing up the calculation for vertical and horizontal asymptotes
  - Correct: Vertical comes from denominator: $x=-\frac{d}{c}$, horizontal from leading coefficients: $y=\frac{a}{c}$
- **Wrong:** Claiming a graph can cross a vertical asymptote
  - Why it fails: Misunderstanding the definition of an asymptote
  - Correct: Vertical asymptotes are lines where the function is undefined, the graph never crosses them

## Cheatsheet

| Feature | Reciprocal $\frac{k}{x-h} + v$ | Linear-over-linear $\frac{ax+b}{cx+d}$ |
| --- | --- | --- |
| Vertical Asymptote | $x = h$ | $x = -\frac{d}{c}$ |
| Horizontal Asymptote | $y = v$ | $y = \frac{a}{c}$ |
| Domain | $x \neq h$ | $x \neq -\frac{d}{c}$ |
| Range | $y \neq v$ | $y \neq \frac{a}{c}$ |
| x-intercept | Solve $\frac{k}{x-h} + v=0$ | $x = -\frac{b}{a}$ |
| y-intercept | Substitute $x=0$ | $y = \frac{b}{d}$ |

## What's next

Reciprocal and rational functions are foundational for more advanced topics in the IB AA SL course. You will apply your knowledge of asymptotes and domain restrictions when you study transformations of functions and solve rational inequalities. These functions also appear regularly in calculus topics, including differentiation and integration of non-polynomial functions, and understanding their key features will help you solve application problems in exams. Mastering this sub-topic also prepares you for working with other non-linear functions like logarithms and trigonometric functions.

- [Exponential and logarithmic functions and graphs](https://www.owlsprep.com/study/ib-math-aa-sl-u2-exponential-and-logarithmic-functions-and/)
- [Geometry & Trigonometry](https://www.owlsprep.com/study/ib-math-aa-sl-u3-overview/)
- [2D and 3D coordinate geometry](https://www.owlsprep.com/study/ib-math-aa-sl-u3-2d-and-3d-coordinate-geometry/)

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