IB Maths Rational Functions: FAQs on the Concept & Exam

According to the IB Mathematics: Analysis and Approaches guide (first exams 2021, current for 2025 syllabus cohorts), rational functions of the form $f(x)=\frac{ax+b}{cx+d}$ appear explicitly at SL, while more general forms (including those with quadratic denominators) are reserved for HL. Command terms to watch for: 'sketch' (rough shape, key features labelled) versus 'draw' (accurate, often on graph paper or GDC-generated).

Quick tip: if you see $x$ in a denominator anywhere in the question, immediately ask 'what value makes this zero?' — that's your vertical asymptote or excluded domain value before you've done anything else.

Worked example: For $f(x)=\frac{2x+3}{x-1}$:

1. Vertical asymptote: set $x-1=0$, so $x=1$.
2. Horizontal asymptote: divide leading coefficients, $y=2/1=2$.
3. Check with a value far from the asymptote, e.g. $x=100$: $f(100)=203/99\approx 2.05$ — confirms it's approaching $y=2$.

HL students also meet oblique (slant) asymptotes when the numerator's degree is exactly one more than the denominator's — found by polynomial long division.

Common mistake: drawing the curve crossing its own horizontal asymptote when it shouldn't, or drawing it as a smooth continuous line straight through the vertical asymptote. For $f(x)=\frac{ax+b}{cx+d}$ the two branches never touch either asymptote line — sketch them curving away instead.

In AI, expect calculator-heavy questions: 'use your GDC to find the equations of the asymptotes.' In AA, expect algebraic derivation without a calculator on Paper 1, then GDC verification on Paper 2 — so you need both skills, not just one.

The IB's Mathematics guide asks for genuine mathematical engagement in the exploration (assessed under Criterion C, personal engagement, and Criterion E, use of mathematics). A rational-function model works well precisely because the asymptote gives you something real to interpret — a maximum yield, a floor cost — rather than just algebra for its own sake.

It shows up more reliably as one part of a multi-part question than as its own dedicated question — so practise spotting it embedded inside longer functions or calculus problems, not just in isolation.

Quick checklist before you submit any rational function answer:

1. Have I stated the excluded value in the domain?
2. Are both asymptotes written as equations ($x=...$, $y=...$), not just marked on the graph?
3. Does my sketch avoid crossing the horizontal asymptote (unless the question genuinely requires it)?
4. Have I checked the y-intercept by substituting $x=0$?

Worked example: Find the inverse of $f(x)=\frac{2x+1}{x-3}$.

1. Write $y=\frac{2x+1}{x-3}$, swap: $x=\frac{2y+1}{y-3}$.
2. Multiply out: $x(y-3)=2y+1$, so $xy-3x=2y+1$.
3. Collect y-terms: $xy-2y=3x+1$, so $y(x-2)=3x+1$.
4. Result: $f^{-1}(x)=\frac{3x+1}{x-2}$.

Notice the coefficient pattern flips — a quick sanity check examiners like to see referenced in your working.

On RevisionPrep, students working through this topic tend to use the Topical Worksheets to isolate rational-function questions specifically, then move to timed Mock Papers once the algebra is automatic — that ordering matters more than volume of practice.

What actually matters for cost-effectiveness: does the resource separate SL from HL content, and does it include full worked solutions (not just final answers)? Those two features are what turn a resource from 'more practice' into genuinely faster learning.