Grade 11 · Precalculus

Rational
Functions

20 essential problems covering key concepts, tricky traps, and everything you need to master rational expressions.

20
Problems
6
Topics
★
Memory Points

§1 · Simplifying Rational Expressions

Factor completely, then cancel common factors — but never forget the restriction!

💡
⚡ Quick Memory Point
FACTOR → CANCEL → RESTRICT
Step 1: Factor numerator & denominator fully.
Step 2: Cancel identical factors.
Step 3: State restrictions (values where denominator = 0 before cancelling).
Factor First Cancel Common State Restrictions Hole ≠ Asymptote
Q 01 Easy Simplifying
📘 Example
Simplify \(\dfrac{x^2-4}{x-2}\).
Factor: \(\dfrac{(x-2)(x+2)}{x-2} = x+2\), where \(x \neq 2\).
Simplify the rational expression: \[\frac{x^2 - 9}{x^2 - x - 6}\] ⚠ Trap: Don't cancel \(x^2\) terms — factor completely first!
Q 02 Easy Simplifying
Simplify: \[\frac{2x^2 + 6x}{4x^2 - 12x}\] ⚠ Trap: Factor out the GCF from both numerator and denominator first!
Q 03 Medium Simplifying
Which expression is equivalent to \(\dfrac{x^2 - 5x + 6}{x^2 - 4}\)? ⚠ Trap: Difference of squares in the denominator — factor as \((x-2)(x+2)\)!

§2 · Asymptotes & Holes

The most commonly confused topic — vertical asymptote vs. hole vs. x-intercept.

🧲
⚡ Quick Memory Point
VA vs HOLE rule:
After full factoring: if a factor cancels → Hole (removable discontinuity).
If a factor stays in denominator only → Vertical Asymptote.

Horizontal Asymptote (HA):
• deg(num) < deg(den) → HA: \(y=0\)
• deg(num) = deg(den) → HA: \(y = \dfrac{\text{leading coeff top}}{\text{leading coeff bottom}}\)
• deg(num) > deg(den) → No HA (oblique asymptote)
Cancel → Hole Stays → VA Bowtie Rule deg< → y=0 deg= → ratio
Q 04 Easy Vertical Asymptote
📘 Example
\(f(x)=\dfrac{x+1}{x-3}\) has VA at \(x=3\) and HA at \(y=1\) (degrees equal, ratio of leading coefficients = 1/1).
What are the vertical asymptote(s) of \(f(x) = \dfrac{3x+1}{x^2-x-6}\)?
Q 05 Medium Hole vs. VA
Consider \(f(x) = \dfrac{x^2 - x - 6}{x^2 - 5x + 6}\). Which statement is correct? ⚠ Most common mistake: confusing a hole with a vertical asymptote.
Q 06 Easy Horizontal Asymptote
What is the horizontal asymptote of \(f(x) = \dfrac{4x^2 - 1}{2x^2 + 3x}\)?
Q 07 Medium Oblique Asymptote
Find the oblique (slant) asymptote of \(f(x) = \dfrac{x^2 - 3x + 2}{x - 1}\). ⚠ Trap: Always simplify (cancel) first before doing polynomial division!

§3 · Domain & Range

Rational functions: domain excludes zeros of the denominator (before simplifying).

🎯
⚡ Quick Memory Point
DOMAIN: Set denominator ≠ 0. Always use original (unsimplified) denominator.
RANGE: Think about y-values the function can never reach — typically excludes the HA value (unless the graph crosses HA).
Denom ≠ 0 Orig Denom Range excludes HA
Q 08 Easy Domain
What is the domain of \(f(x) = \dfrac{x+2}{x^2 - 4x + 3}\)?
Q 09 Medium Domain with Hole
\(g(x) = \dfrac{x^2 - 4}{x^2 + x - 6}\). Which is the correct domain? ⚠ Trap: Use the original denominator — don't cancel first, then find domain!

§4 · Multiplying & Dividing

✖️
⚡ Quick Memory Point
Multiply: Factor everything → cross-cancel → multiply straight across.
Divide: KCF — Keep the first, Change to multiply, Flip the second.
KCF (Keep·Change·Flip) Cross-Cancel Factor First
Q 10 Easy Multiplying
📘 Example
\(\dfrac{x}{x+1} \cdot \dfrac{x+1}{x-2} = \dfrac{x}{x-2}\) (cancel \(x+1\)), where \(x \neq -1, 2\).
Simplify: \(\dfrac{x^2-1}{x+3} \cdot \dfrac{x^2-9}{x-1}\)
Q 11 Medium Dividing
Simplify: \(\dfrac{x^2-4}{x^2+x} \div \dfrac{x-2}{x^2+2x+1}\) ⚠ Trap: Flip the SECOND fraction only — never flip the first!

§5 · Adding & Subtracting

➕
⚡ Quick Memory Point
LCD Rule: Find LCD → rewrite each fraction → add/subtract numerators → simplify result.
Subtraction trap: Distribute the negative sign across the entire numerator!
\(\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad - bc}{bd}\) — the minus sign belongs to the whole second numerator.
Find LCD Distribute Minus Simplify Result
Q 12 Easy Adding
📘 Example
\(\dfrac{2}{x} + \dfrac{3}{x+1}\): LCD = \(x(x+1)\).
\(= \dfrac{2(x+1)+3x}{x(x+1)} = \dfrac{5x+2}{x(x+1)}\)
Add: \(\dfrac{3}{x-2} + \dfrac{5}{x+2}\)
Q 13 Medium Subtracting
Subtract: \(\dfrac{4x}{x^2-1} - \dfrac{2}{x+1}\) ⚠ Trap: Factor \(x^2-1 = (x-1)(x+1)\) first. LCD = \((x-1)(x+1)\).
Q 14 Medium Complex Fractions
Simplify the complex fraction: \(\dfrac{\dfrac{1}{x} - \dfrac{1}{y}}{\dfrac{1}{x} + \dfrac{1}{y}}\) ⚠ Trick: Multiply top and bottom by LCD = \(xy\).

§6 · Solving Rational Equations

Clear the denominator — then always check for extraneous solutions!

🔍
⚡ Quick Memory Point
MULTIPLY BOTH SIDES by LCD → solve → CHECK!
An extraneous solution makes a denominator = 0 in the original equation. Always substitute your answer back!
Multiply by LCD Check Extraneous Sub Back In
Q 15 Easy Solving
📘 Example
Solve \(\dfrac{2}{x} = \dfrac{3}{x+1}\).
Multiply both sides by \(x(x+1)\): \(2(x+1)=3x \Rightarrow 2x+2=3x \Rightarrow x=2\). ✓
Solve: \(\dfrac{5}{x-3} = \dfrac{2}{x-1}\)
Q 16 Hard Extraneous Solutions
Solve: \(\dfrac{x}{x-2} + \dfrac{1}{x+2} = \dfrac{8}{x^2-4}\) ⚠ Watch carefully — one solution may be extraneous!
Q 17 Medium Solving
Solve: \(\dfrac{3}{x+1} - \dfrac{1}{x-1} = \dfrac{2}{x^2-1}\)
Q 18 Medium Inequalities
Solve the rational inequality: \(\dfrac{x+1}{x-3} > 0\) ⚠ Sign chart method: find critical values, test each interval. Never multiply both sides by \((x-3)\) without checking its sign!
Q 19 Hard Graph Behaviour
The function \(f(x) = \dfrac{2x^2 + x - 1}{x^2 - 1}\) has which features? ⚠ Factor completely first — there may be holes you don't expect!
Q 20 Hard Mixed Mastery
Which rational function has a hole at \(x=1\), a vertical asymptote at \(x=-2\), and a horizontal asymptote at \(y=1\)?

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