Part 1 of 7 โ Definition, Domain & Excluded Values
A rational function is a ratio of two polynomials:
f(x)=q(x)p(x)โ,q(x)๎ =0โ
Just as you can't divide numbers by zero, you can't divide polynomials by zero. This makes the domain โ the set of all legal inputs โ the first thing to determine when working with any rational function.
๐ Recognizing Rational Functions
Expression
Rational?
Why
xโ3x2+1โ
โ
Polynomial over polynomial
๐ Finding the Domain
The domain of f(x)=q(x)p(x)โ is all real numbers except where .
Domain & Excluded Values Quiz ๐ฏ
Domain Drill ๐งฎ
1) For f(x)=xโ67โ, what value of x is excluded from the domain? (e.g., for , set to get )
Domain Concepts โ Fill in the Blanks ๐ฝ
Exit Quiz โ Domain & Excluded Values โ
Part 2: Vertical Asymptotes
๐ Vertical & Horizontal Asymptotes
Part 2 of 7 โ Predicting Long-Run and Singular Behavior
Asymptotes are invisible boundary lines that a rational function's graph approaches but (usually) never reaches. They tell us what happens at the extremes โ near excluded values and as xโยฑโ.
๐ Vertical Asymptotes
A vertical asymptote occurs at x=c when:
(denominator is zero), AND
Part 3: Horizontal & Slant Asymptotes
๐ณ๏ธ Holes & Removable Discontinuities
Part 3 of 7 โ When Factors Cancel
Not every denominator zero produces a vertical asymptote. When numerator and denominator share a common factor, canceling it creates a hole โ a single missing point where the function is undefined but the graph has no dramatic blow-up.
๐ Holes vs. Vertical Asymptotes
When q(c)=0, there are two possibilities:
Situation
(xโc) cancels?
Part 4: Graphing Rational Functions
โ๏ธ Simplifying Rational Expressions
Part 4 of 7 โ Algebraic Simplification, Addition, & Division
Before you can graph or analyze a rational function, you often need to simplify it first. This part covers the algebraic mechanics: factoring and canceling, adding/subtracting with common denominators, multiplying/dividing rational expressions, and rewriting via polynomial long division.
๐ Factoring & Canceling
The fundamental simplification technique:
Bโ C
Part 5: Solving Rational Equations
๐ Graphing Rational Functions
Part 5 of 7 โ Transformations & Complete Graph Sketching
Graphing a rational function means assembling all the pieces from Parts 1โ4: domain, intercepts, asymptotes, holes, and sign behavior. This part gives you a systematic graphing procedure and introduces transformations of the parent function y=x1โ.
๐ The Parent Function
Part 6: Problem-Solving Workshop
โ๏ธ Rational Equations & Inequalities
Part 6 of 7 โ Solving Rational Equations, Checking for Extraneous Solutions, and Rational Inequalities
Up to now we have analyzed rational functions. This part shifts to solving โ finding x-values that satisfy rational equations and inequalities. The critical new skill is checking for extraneous solutions introduced when you multiply both sides by an expression containing the variable.
๐ Solving Rational Equations
The LCD Method
Step
Action
Example: x
Part 7: Review & Applications
๐ Rational Functions โ Full Synthesis
Part 7 of 7 โ Putting It All Together
This final part integrates every concept: domain, asymptotes, holes, simplification, graphing, and solving. The problems are multi-step โ just like exam questions.
Your Rational Functions Toolkit
Concept (Part)
Key Question
Domain & Excluded Values (1)
Where is the denominator zero?
Vertical & Horizontal Asymptotes (2)
What happens near excluded values and at ยฑโ?
Holes (3)
Do any factors cancel?
Simplification (4)
Can we factor, combine, or divide?
Graphing (5)
What does the complete picture look like?
Equations & Inequalities (6)
What x-values satisfy the condition?
x5โ
โ
Constant (degree 0) over polynomial
x+1xโโ
โ
Numerator is not a polynomial
1x3โ2xโ
โ
Any polynomial is rational (q(x)=1)
x2+2x+13xโ
โ
Polynomial over polynomial
๐ก Every polynomial is also a rational function โ just one with denominator 1.
q(x)=0
Step-by-Step Process
Step
Action
Example: f(x)=x2โ4x+1โ
1
Set denominator equal to zero
x2โ4=0
2
Solve for x
Worked Examples
Example 1:g(x)=x+53xโ
Set x+5=0โนx=โ5
Domain:ย (โโ,โ5)โช(โ5,โ)
Example 2:h(x)=x2+1x2โ
Set x2+1=0โนx2=โ1 โ no real solutions!
Domain:ย (โโ,โ)=allย realย numbers
โ ๏ธ Not every rational function has excluded values. If the denominator has no real roots, the domain is all reals.
2) For g(x)=2x+10xโ, what value of x is excluded? (e.g., for 3x+12xโ, set 3x+12=0 to get x=โ4)
3) How many values are excluded from the domain of h(x)=x2+51โ? (e.g., for x2+11โ, x2+1=0 has no real solutions โ 0 excluded)
q
(
c
)
=
0
The factor (xโc) does not cancel with the numerator
VAย atย x=cโบq(c)=0ย andย p(c)๎ =0โ
What Happens Near a VA
As x approaches c, f(x)โ+โ or f(x)โโโ (the graph shoots up or down).
Worked Example
Find the vertical asymptote(s) of f(x)=x2โ12xโ.
Step 1: Factor denominator: x2โ1=(xโ1)(x+1)
Step 2: Set each factor to zero: x=1 and x=โ1
Step 3: Check numerator: 2(1)=2๎ =0 and 2(โ1)=โ2๎ =0
Result: Vertical asymptotes at x=1 and x=โ1
โ ๏ธ If both numerator and denominator are zero at x=c, the common factor cancels and you get a hole (Part 3), not a vertical asymptote.
๐ Horizontal Asymptotes
A horizontal asymptote tells you the output value that f(x) approaches as xโยฑโ. It depends entirely on comparing the degrees of the numerator and denominator.
Degree Comparison
Horizontal Asymptote
Why
deg(p)<deg(q)
y=0
Denominator grows faster โ ratio shrinks to 0
deg(p)=deg(q)
y=b (ratio of leading coefficients)
deg(p)>deg(q)
None (oblique/slant asymptote instead)
Numerator grows faster โ ratio grows without bound
Worked Examples
Example 1:f(x)=x2+53x+1โ โ โ HA:
Example 2:g(x)=2x2+34x โ โ HA:
Example 3:h(x)=x+1x3โ โ โ No HA (slant asymptote exists)
๐ก Memory aid: "Bottom wins โ y=0. Tie โ ratio of leaders. Top wins โ no HA."
๐ Slant (Oblique) Asymptotes
When deg(p)=deg(q)+1 (numerator is exactly one degree higher), the function has a slant asymptote found by polynomial long division.
Example
Find the slant asymptote of f(x)=x+1x2+3x+5โ.
Divide x2+3x+5 by x+1:
x2+3x+5=(x+1)(x+2)+
f(x)=x+2+x+13โ
As xโยฑโ, x+13โโ0, so:
Slantย asymptote:ย y=x+2โ
Asymptote Quiz ๐ฏ
Asymptote Drill ๐งฎ
1) What is the horizontal asymptote of f(x)=3x2โ76x2+1โ? Give the y-value. (e.g., for 4x2+18x2โ, HA is y=48โ=2)
2) How many vertical asymptotes does g(x)=x2โ5x+61โ have? (e.g., for , factor to โ VAs)
3) For h(x)=x+1x2+xโ, after canceling the common factor, what is simplified? Give just the simplified expression as a number (evaluate ). (e.g., , so )
โ ๏ธ Never cancel terms โ only factors!x+5x+3โ๎ =53โ. You can only cancel something that multiplies the entire numerator and entire denominator.
๐ง Operations with Rational Expressions
Adding & Subtracting (LCD Method)
BAโ+DCโ=BDAD+BCโ
Example:xโ12โ+x+43โ
LCD =(xโ1)(x+4)
=
Multiplying & Dividing
Operation
Rule
Example
Multiply
BAโโ D
๐ก Always factor before multiplying โ it makes cancellation much easier.
โ๏ธ Polynomial Long Division for Rationals
When deg(p)โฅdeg(q), you can rewrite q(x)p(x)โ as:
q(x)p(x)โ=quotient+
This is essential for finding slant asymptotes and understanding end behavior.
Worked Example
Rewrite x+22x2+3xโ5โ in quotient-remainder form.
Dividing:
2x2รทx=2x. Multiply: 2x(x+2)=. Subtract: .
x+22x2+
As xโยฑโ, x+2โ3โโ0, so the slant asymptote is .
Simplification Quiz ๐ฏ
Simplification Drill ๐งฎ
1) Simplify x+5x2โ25โ and evaluate at x=3. (e.g., x+2x2โ4โ=xโ2, so at x=3: 3โ2=1)
2) What is x2โ+x3โ? Evaluate at . (e.g., , so at : )
3) Divide: x+1x2+2x+1โ. Evaluate at x. (e.g., , so at : )
Simplification Rules โ Fill in the Blanks ๐ฝ
Exit Quiz โ Simplification โ
y
=
x1โ
Every simple rational function is a transformation of this parent graph.
Feature
Value
Domain
(โโ,0)โช(0,โ)
Range
(โโ,0)โช(0,โ)
VA
x=0
HA
y=0
Symmetry
Odd function (symmetric about the origin)
Quadrants
I and III
Transformation Form
f(x)=xโhaโ+kโ
Parameter
Effect
Example
h
Shifts graph righth units (VA moves to x=h)
h=3: VA at x=3
k
Shifts graph upk units (HA moves to y=k)
k=โ2: HA at
a
Vertical stretch by $
a
Example:f(x)=x+1โ2โ+3 has VA at x=โ1, HA at y=3, reflected and stretched by 2.
๐ Complete Graphing Procedure
Follow these steps for any rational function f(x)=q(x)p(x)โ:
Step
Action
What It Gives You
1
Factor numerator and denominator completely
Reveals all features at once
2
Find domain exclusions
Where q(x)=0
3
Identify holes (common factors)
Points to mark with open circles
4
Find vertical asymptotes (remaining denom zeros)
Dashed vertical lines
5
Find horizontal/slant asymptote
Dashed horizontal or diagonal line
6
Find x-intercepts
Set (after canceling)
Worked Example
Sketch f(x)=x2โ4xโ1โ
Factor:(xโ2)(x+2)xโ1โ โ no common factors
Domain:
Graphing Quiz ๐ฏ
Graphing Features Drill ๐งฎ
1) What is the y-intercept of f(x)=xโ1x+3โ? Give the y-value. (e.g., for xโ4x+2โ, f(0)=โ42โ=โ0.5)
2) For g(x)=x+45โโ2, what is the horizontal asymptote? Give the y-value. (e.g., has HA at )
3) How many vertical asymptotes does h(x)=x3โxxโ have after simplification? (e.g., has VA)
Graphing Concepts โ Fill in the Blanks ๐ฝ
Exit Quiz โ Graphing โ
3
โ
+
x+21โ=
1
1
Find the LCD
LCD=x(x+2)
2
Multiply every term by the LCD
3(x+2)+x=x(x+2)
3
Expand and simplify
3x+6+x=x2+2x
4
Collect to one side
x2โ2xโ6=0
5
Solve (quadratic formula)
x=22ยฑ4+24
6
Check for extraneous solutions
Neither value makes x=0 or x=โ2 โ
โ ๏ธ Step 6 is mandatory. Multiplying by the LCD can introduce false solutions that make the original denominator zero.
๐จ Extraneous Solutions
What Are They?
An extraneous solution is a value that satisfies the transformed equation but makes a denominator in the original equation equal to zero.
โฅ0: want positive or zero. Include x=1 (zero), exclude x=โ3 (undefined).
(โโ,โ3)โช[1,โ)โ
Equations & Inequalities Quiz ๐ฏ
Solving Drill ๐งฎ
1) Solve: x6โ=2. What is x? (e.g., x10โ=5 โ x=510โ=2)
2) How many extraneous solutions arise when solving xโ2x2โ4โ=x+2? (e.g., if the only solution makes a denominator zero, that is extraneous solution)
3) For xโ3x+1โ>0, how many intervals are in the solution set? (e.g., has solution โ that is intervals)