IB Maths AA HLTopic 2 — FunctionsPaper 1 & 2~9 min read
Reciprocal & Rational Functions
A rational function is just a fraction whose top and bottom are both polynomials. The simplest is y = 1/x — the reciprocal function. The general linear-over-linear form (ax + b)/(cx + d) builds on it by stretching, shifting, and reflecting that basic shape. The four things you’ll always need to find: y-intercept, x-intercept, vertical asymptote, horizontal asymptote. Once you have these four, sketching the graph is automatic.
📘 What you need to know
Reciprocal function: f(x) = 1/x, with x ≠ 0. Domain and range both ℝ \ {0}. Self-inverse: f−1(x) = f(x).
Reciprocal graph: two asymptotes (x = 0 and y = 0); two axes of symmetry (y = x and y = −x); no intercepts; no max/min; two branches in opposite quadrants.
Linear rational function: f(x) = (ax + b)/(cx + d), with x ≠ −d/c.
Y-intercept: substitute x = 0 → b/d. X-intercept: solve numerator = 0 → x = −b/a.
Vertical asymptote: solve denominator = 0 → x = −d/c.
Horizontal asymptote: y = a/c (ratio of leading coefficients — what the function tends to as x → ±∞).
Domain: ℝ except x = −d/c. Range: ℝ except y = a/c.
The inverse is also a rational function — find it by the swap-and-rearrange method (no need to memorise a formula).
Apply the function twice: f(f(x)) = 1/(1/x) = x. Flipping a number, then flipping again, gets you back to the original. So 1/x is its own inverse — it equals its mirror image in y = x.
Linear rational functions — (ax + b)/(cx + d)
Linear rational functionf(x) = ax + bcx + d, x ≠ −d/c
This is the general linear-over-linear shape — same two-branch structure as 1/x, but shifted so the asymptotes aren’t on the axes anymore. The four key features come from simple substitutions:
y-intercept
y = b/d
substitute x = 0
x-intercept
x = −b/a
solve numerator = 0
Vertical asymptote
x = −d/c
solve denominator = 0
Horizontal asymptote
y = a/c
ratio of leading coefficients
A general (ax + b)/(cx + d) graph with all four features
Memorise the rule “denominator = 0 → vertical asymptote; ratio of leading coefficients → horizontal asymptote“. This works for every rational function in this section.
How to sketch a linear rational function
🧭 Recipe — sketching y = (ax + b)/(cx + d)
Find the y-intercept: substitute x = 0 → coordinate (0, b/d).
Find the x-intercept: set numerator = 0 → coordinate (−b/a, 0).
Find the vertical asymptote: set denominator = 0 → vertical line x = −d/c.
Find the horizontal asymptote: y = a/c — horizontal line.
Sketch: draw the asymptotes as dashed lines; mark the intercepts; draw two branches that approach the asymptotes without crossing them.
Label everything on your sketch — examiners check for asymptote equations and intercept coordinates.
Domain, range, and the inverse
Domain & range
domain: x ≠ −d/c range: y ≠ a/c
excludes the vertical asymptote x-value; range excludes the horizontal asymptote y-value
Inverse
also a rational function
find by swap-and-rearrange (no formula to memorise)
Reflection check: the graph of f−1(x) is the reflection of f(x) in the line y = x. So vertical and horizontal asymptotes swap when you take the inverse.
Worked examples
WE 1
Find the four key features of a linear rational function
For f(x) = 3x − 6x + 2, x ≠ −2, find: (a) the y-intercept, (b) the x-intercept, (c) the vertical asymptote, (d) the horizontal asymptote.
(a) y-intercept: substitute x = 0f(0) = (0 − 6)/(0 + 2) = −6/2 = −3 → (0, −3)(b) x-intercept: numerator = 03x − 6 = 0 → x = 2 → (2, 0)(c) Vertical asymptote: denominator = 0x + 2 = 0 → x = −2(d) Horizontal asymptote: ratio a/ca = 3, c = 1 → y = 3y-int (0, −3); x-int (2, 0); VA x = −2; HA y = 3always do all four — the marks come from labelling, not algebra
WE 2
Sketch a rational function with all features labelled
Sketch the graph of f(x) = 2x + 4x − 1, marking the intercepts and asymptotes.
Find all four featuresy-int: f(0) = 4/(−1) = −4 → (0, −4)x-int: 2x + 4 = 0 → x = −2 → (−2, 0)VA: x − 1 = 0 → x = 1HA: y = 2/1 = 2Sketchdraw asymptotes x = 1 and y = 2 as dashed linesleft branch passes through (−2, 0) and (0, −4), approaches both asymptotesright branch sits in upper-right region, also approaching bothsketch with VA x = 1, HA y = 2, x-int (−2, 0), y-int (0, −4)the two branches sit on opposite sides of where VA and HA meet — visualise that intersection point as the “centre”
WE 3
Find the inverse of a linear rational function
Find f−1(x) for f(x) = x + 4x − 3 and state its domain.
Step 1: Set y = f(x) and swap x and yy = (x + 4)/(x − 3)swap → x = (y + 4)/(y − 3)Step 2: Multiply by (y − 3) and rearrangex(y − 3) = y + 4xy − 3x = y + 4xy − y = 3x + 4y(x − 1) = 3x + 4y = (3x + 4)/(x − 1)Step 3: Domain of f⁻¹ = range of f → exclude HA y = 1f⁻¹(x) = (3x + 4)/(x − 1), x ≠ 1note: HA of f was y = 1 (a/c = 1/1) — that becomes the excluded x-value for f⁻¹
WE 4
State domain and range
Find the domain and range of f(x) = 5x − 12x + 3.
Domain: exclude denominator = 02x + 3 = 0 → x = −3/2domain: x ∈ ℝ, x ≠ −3/2Range: exclude horizontal asymptoteHA: y = 5/2range: f(x) ∈ ℝ, f(x) ≠ 5/2domain: x ≠ −3/2; range: f(x) ≠ 5/2remember: domain excludes VA x-value; range excludes HA y-value
WE 5
Show 1/x is self-inverse
Show that f(x) = 1/x, x ≠ 0 is self-inverse, and find f(f(7)).
Compute f(f(x))f(f(x)) = f(1/x) = 1 / (1/x) = x ✓so f⁻¹ = f → self-inverseApply at x = 7f(7) = 1/7, then f(1/7) = 7f is self-inverse; f(f(7)) = 7flip, flip, back to start — works for any non-zero input
WE 6
Identify a rational function from its features
A rational function of the form f(x) = (ax + b)/(x + d) has vertical asymptote x = 4, horizontal asymptote y = 3, and y-intercept (0, −2). Find a, b, and d.
Step 1: VA gives dx = 4 means denominator zero at x = 4 → x + d = 0 → d = −4Step 2: HA gives ay = a/c, with c = 1 (coefficient of x in denom) → a = 3Step 3: y-intercept gives bf(0) = b/d = b/(−4) = −2b = 8a = 3, b = 8, d = −4 → f(x) = (3x + 8)/(x − 4)three features, three unknowns — work them out one at a time using the right-side relationships
💡 Top tips
The four features are the whole game: y-intercept (substitute x = 0); x-intercept (numerator = 0); VA (denominator = 0); HA (ratio of leading coefficients).
Asymptotes are dashed on a sketch. The curve approaches but doesn’t cross them.
Always label asymptote equations (x = …, y = …) and intercept coordinates on your sketch.
For domain: exclude any x that makes the denominator zero.
For range: exclude the horizontal asymptote y-value.
1/x is self-inverse — and so is any function of the form (ax + b)/(cx − a) (where the constant in denom is the negative of a).
Sketch with the GDC first if you’re unsure — then transfer the key features to your hand sketch.
⚠ Common mistakes
Drawing asymptotes as solid lines. They should always be dashed — the curve doesn’t actually touch them.
Forgetting to label intercepts on the sketch — examiners deduct marks for incomplete sketches.
Confusing horizontal asymptote rule: it’s the ratio of leading coefficients (a/c), not the ratio of constants.
Using b/d for the x-intercept. The y-intercept is b/d; the x-intercept is −b/a.
Sketching the curve crossing the asymptote. Linear-over-linear graphs never touch their asymptotes.
Forgetting the range excludes the horizontal asymptote. Domain ≠ range.
Not stating the domain restriction when finding an inverse — the inverse has its own VA-related exclusion.
The next note steps up the complexity to rational functions with quadratics — fractions with quadratics on top, on bottom, or both. The number of asymptotes can change (you may get two, one, or none), and you’ll meet a new type — oblique asymptotes — which require a polynomial division to find.
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