Consider the function $f(x) = \frac{5x - 2}{x + 3}$ , where $x \ne -3$ . Find an expression for the inverse function, $f^{-1}(x)$ . State the equations of the vertical and horizontal asymptotes of $f^{-1}(x)$ .

Let $y = \frac{5x - 2}{x + 3}$ . Swap $x$ and $y$ : $x = \frac{5y - 2}{y + 3}$ . M1 Multiply out: $xy + 3x = 5y - 2$ . Collect $y$ terms: $xy - 5y = -3x - 2 \implies y(x - 5) = -3x - 2$ . (A1) $f^{-1}(x) = \frac{-3x - 2}{x - 5}$ (or $\frac{3x+2}{5-x}$ ). A1 Vertical asymptote: $x = 5$ . A1 Horizontal asymptote: $y = -3$ . A1 (Note: these are the asymptotes $y=5$ and $x=-3$ of $f(x)$ , reflected in $y=x$ .)

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IB Maths AA SL · Unit 2: Functions

IB Maths AA SL Rational and Reciprocal Functions Questions

Updated October 2026

Exam-style IB Maths AA SL rational and reciprocal functions questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

45 questions
Paper 1: 38
Paper 2: 7
8 easy
23 medium
10 hard
4 very hard
3 worked examples

Practise Rational and Reciprocal Functions questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

Vertical asymptotes from the denominator = 0; horizontal from the ratio of leading terms as x → ∞. Sketch the shape, don't rely on a GDC — Paper 1 is calc-free. Rational functions and asymptotes overview →

What's examined in AA SL rational and reciprocal functions

The question bank covers these rational and reciprocal functions question types (number of questions in brackets):

Rational Function Properties (31)
Solving Rational Equations and Graphs (9)
Inverse and Composite Rational Functions (5)

Where students lose marks on rational and reciprocal functions

Examiner's noteAnswer in the wrong form: asymptotes not as equations, coordinates vs values, unfinished answers

Where marks are lost: A recurring loss of easy marks: an asymptote written as "HA = …", "y ≠ …" or just a number instead of an equation y = …; coordinates given when values of x were asked for (or the reverse); an axis of symmetry not written as x = …; interval answers in the wrong notation; and answers left unfinished, such as an unevaluated logarithm, a fraction of decimals or a surd that still simplifies.

What examiners look for: Lines are equations: write y = … or x = …. Give exactly what the question names (a value, a point, an interval). Finish the arithmetic: integers as integers, fractions with integer numerators and denominators in lowest terms.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Rational and Reciprocal Functions worked examples

Worked example 1: Finding asymptotes and intercepts · easy

Let $f (x) = \frac{4 x - 5}{2 x + 3}$ . Find the exact equations of both asymptotes and the exact coordinates of the intercepts.

Solution

4 steps

Examiner tip: You can verify with a GDC graph, but exact values are best (and fastest) found algebraically.

Worked example 2: Inverse of a rational function · medium

Find the inverse function $f^{- 1} (x)$ for $f (x) = \frac{3 x + 4}{2 x - 1}$ , where $x \neq 0.5$ .

Solution

5 steps

Examiner tip: If you end up with all-negative signs (e.g.

\frac{- x - 4}{3 - 2 x}

), multiply top and bottom by

- 1

to match the examiner's mark scheme presentation.

Worked example 3: Partial fractions and translating reciprocals · hard

Let $f (x) = \frac{3 x + 5}{x + 2}$ for $x \neq - 2$ . Show that $f (x)$ can be written as $A + \frac{B}{x + 2}$ , and describe the transformations mapping $y = \frac{1}{x}$ onto $y = f (x)$ .

Solution

6 steps · 1 to fill in

Examiner tip: Rewriting the numerator as a multiple of the denominator (e.g.

3 x + 5 = 3 (x + 2) - 1

) is safer than formal polynomial long division, which is prone to sign errors.

Try these IB Maths AA SL rational and reciprocal functions questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 5 marks · Paper 1

The function $f$ is defined by $f (x) = \frac{1}{x - 2} + 3$ , for $x \neq 2$ .

Write down the equations of the vertical and horizontal asymptotes of the graph of $f$ .
Find the exact coordinates of the $y$ -intercept.
Find the exact coordinates of the $x$ -intercept algebraically.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Consider the function $f (x) = \frac{5 x - 2}{x + 3}$ , where $x \neq - 3$ .

Find an expression for the inverse function, $f^{- 1} (x)$ .
State the equations of the vertical and horizontal asymptotes of $f^{- 1} (x)$ .

Full worked solution Free, no sign-in

Let $y = \frac{5 x - 2}{x + 3}$ . Swap $x$ and $y$ : $x = \frac{5 y - 2}{y + 3}$ . M1
Multiply out: $x y + 3 x = 5 y - 2$ .
Collect $y$ terms: $x y - 5 y = - 3 x - 2 ⟹ y (x - 5) = - 3 x - 2$ . (A1)
$f^{- 1} (x) = \frac{- 3 x - 2}{x - 5}$ (or $\frac{3 x + 2}{5 - x}$ ). A1
Vertical asymptote: $x = 5$ . A1
Horizontal asymptote: $y = - 3$ . A1
(Note: these are the asymptotes $y = 5$ and $x = - 3$ of $f (x)$ , reflected in $y = x$ .)

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 1

The curve $C$ has equation $y = \frac{2 x + 1}{x - 1}$ for $x \neq 1$ . The straight line $L$ has equation $y = - x + k$ , where $k$ is a constant.

Show that the $x$ -coordinates of any points of intersection of $C$ and $L$ satisfy the equation $x^{2} + (1 - k) x + (k + 1) = 0$ .
Given that the line $L$ is a tangent to the curve $C$ , use the discriminant to find the exact possible values of $k$ .

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AA SL rational and reciprocal functions questions are there?

There are 45 exam-style rational and reciprocal functions questions in the AA SL question bank (Paper 1: 38 · Paper 2: 7), graded 8 easy, 23 medium, 10 hard, 4 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is rational and reciprocal functions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 38 · Paper 2: 7. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA SL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

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Free ready-to-teach lesson (slides and teacher notes): Rational functions & asymptotes.

IB Maths AA SL Rational and Reciprocal Functions Questions

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What you need to know

What's examined in AA SL rational and reciprocal functions

Where students lose marks on rational and reciprocal functions

Rational and Reciprocal Functions worked examples

Worked example 1: Finding asymptotes and intercepts · easy

Solution

Worked example 2: Inverse of a rational function · medium

Solution

Worked example 3: Partial fractions and translating reciprocals · hard

Solution

Try these IB Maths AA SL rational and reciprocal functions questions

Question 1 · easy · 5 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 1

Revise rational and reciprocal functions for the exam

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