Rational Functions - Formulas & Properties
IB Mathematics Analysis & Approaches (SL & HL)
📐 Definition
General Form:
A rational function is the ratio of two polynomial functions where the denominator is not equal to zero.
\[f(x) = \frac{P(x)}{Q(x)}\]
where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \neq 0\)
🔄 The Reciprocal Function
Definition:
\[f(x) = \frac{1}{x}, \quad x \neq 0\]
Key Properties:
Domain: \(x \in \mathbb{R}, x \neq 0\)
Range: \(y \in \mathbb{R}, y \neq 0\)
Vertical Asymptote: \(x = 0\)
Horizontal Asymptote: \(y = 0\)
Symmetry: Odd function (symmetric about the origin)
Self-Inverse Property:
\[f^{-1}(x) = f(x) = \frac{1}{x}\]
The graph is unchanged when reflected in the line \(y = x\)
📊 Linear Rational Functions
Standard Form:
\[f(x) = \frac{ax + b}{cx + d}\]
where \(a, b, c, d\) are constants and \(c \neq 0\)
Domain:
\[x \in \mathbb{R}, x \neq -\frac{d}{c}\]
All real numbers except where the denominator equals zero
Range:
\[y \in \mathbb{R}, y \neq \frac{a}{c}\]
All real numbers except the horizontal asymptote value
📏 Asymptotes
Vertical Asymptote:
Occurs where the denominator equals zero (and numerator is non-zero at that point).
\[x = -\frac{d}{c}\]
For \(f(x) = \frac{ax + b}{cx + d}\), set \(cx + d = 0\) and solve for \(x\)
Horizontal Asymptote:
Describes the behavior as \(x \to \pm\infty\). For linear rational functions:
\[y = \frac{a}{c}\]
Ratio of leading coefficients when degrees are equal
General Rules for Horizontal Asymptotes:
If degree of numerator = degree of denominator:
\(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\)
If degree of numerator < degree of denominator: \(y = 0\)
If degree of numerator > degree of denominator: No horizontal asymptote (possible oblique asymptote)
⭕ Finding Intercepts
x-intercept (Root):
Set \(f(x) = 0\) and solve. This occurs when the numerator equals zero.
\[ax + b = 0 \implies x = -\frac{b}{a}\]
The function has one x-intercept (provided \(a \neq 0\))
y-intercept:
Set \(x = 0\) and evaluate \(f(0)\).
\[f(0) = \frac{b}{d}\]
The y-intercept is \(\left(0, \frac{b}{d}\right)\) (provided \(d \neq 0\))
🔢 Rational Functions with Quadratics (HL)
Form 1: Linear over Quadratic
\[f(x) = \frac{ax + b}{cx^2 + dx + e}\]
Vertical Asymptotes: Solve \(cx^2 + dx + e = 0\) (up to 2 vertical asymptotes)
Horizontal Asymptote: \(y = 0\) (degree of denominator > numerator)
Form 2: Quadratic over Linear
\[f(x) = \frac{ax^2 + bx + c}{dx + e}\]
Vertical Asymptote: Solve \(dx + e = 0\) (one vertical asymptote)
Oblique Asymptote: Use polynomial long division to find slant asymptote
Form 3: Quadratic over Quadratic
\[f(x) = \frac{ax^2 + bx + c}{dx^2 + ex + f}\]
Vertical Asymptotes: Solve \(dx^2 + ex + f = 0\)
Horizontal Asymptote: \(y = \frac{a}{d}\) (degrees equal)
🔧 Transformations of Reciprocal Function
General Transformation:
\[f(x) = \frac{a}{x - h} + k\]
• \(a\): Vertical stretch/compression and reflection
• \(h\): Horizontal shift (moves vertical asymptote to \(x = h\))
• \(k\): Vertical shift (moves horizontal asymptote to \(y = k\))
Asymptotes After Transformation:
Vertical Asymptote: \(x = h\)
Horizontal Asymptote: \(y = k\)
⭐ Special Properties
No Maximum or Minimum:
Rational functions do not have global maxima or minima (they approach infinity)
Discontinuity:
Rational functions are discontinuous at vertical asymptotes
End Behavior:
As \(x \to \pm\infty\), the function approaches its horizontal asymptote
🔀 Inverse of Linear Rational Function
Finding the Inverse:
For \(f(x) = \frac{ax + b}{cx + d}\), the inverse is also a rational function:
\[f^{-1}(x) = \frac{dx - b}{-cx + a}\]
You can derive this in your exam by swapping \(x\) and \(y\), then solving for \(y\)
✏️ Sketching Rational Functions
Essential Steps:
1. Find vertical asymptote(s): Set denominator = 0
2. Find horizontal asymptote: Compare degrees or use limits
3. Find y-intercept: Calculate \(f(0)\)
4. Find x-intercept(s): Set numerator = 0
5. Plot key points: Choose points on either side of asymptotes
6. Draw asymptotes: Use dashed lines
7. Sketch curves: Draw smooth curves approaching asymptotes
8. Label everything: Coordinates of intercepts and equations of asymptotes
💡 Exam Tip: Always identify asymptotes first when sketching rational functions. Remember that the graph never crosses a vertical asymptote but may cross a horizontal asymptote. Use your GDC to verify your sketch and check behavior near asymptotes.