IB MYP 4-5 Maths- Rational functions- Study Notes - New Syllabus
IB MYP 4-5 Maths- Rational functions – Study Notes
Extended
- Rational functions
IB MYP 4-5 Maths- Rational functions – Study Notes – All topics
Rational Functions
Rational Functions
A rational function is a function that can be written as:
\( f(x) = \dfrac{P(x)}{Q(x)} \)
where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \neq 0 \).
Key Characteristics:
Domain: All real numbers except where \( Q(x) = 0 \) (denominator cannot be zero).
Vertical Asymptotes: Occur where \( Q(x) = 0 \).
Horizontal Asymptotes:
- If degree of numerator < degree of denominator → \( y = 0 \).
- If degrees are equal → \( y = \dfrac{\text{leading coefficient of P}}{\text{leading coefficient of Q}} \).
- If degree of numerator > degree of denominator → no horizontal asymptote (may have oblique asymptote).
Intercepts:
- x-intercept: Solve \( P(x) = 0 \).
- y-intercept: Substitute \( x = 0 \).
Behavior near asymptotes: Function approaches infinity or negative infinity near vertical asymptotes.
General Steps to Analyze a Rational Function:
- Find domain by excluding values where denominator is zero.
- Find vertical asymptotes by solving \( Q(x) = 0 \).
- Find horizontal asymptote using degree rules.
- Find x-intercepts by solving \( P(x) = 0 \).
- Find y-intercept by substituting \( x = 0 \).
Example :
Analyze \( f(x) = \dfrac{1}{x} \).
▶️ Answer/Explanation
- Domain: \( x \neq 0 \).
- Vertical Asymptote: \( x = 0 \).
- Horizontal Asymptote: \( y = 0 \) (degree numerator < denominator).
- Intercepts: None (cannot cross axes).
Graph has two branches in quadrants I and III.
Example :
Analyze \( f(x) = \dfrac{2x + 3}{x – 1} \).
▶️ Answer/Explanation
- Domain: \( x \neq 1 \).
- Vertical Asymptote: \( x = 1 \).
- Horizontal Asymptote: Degrees are equal → \( y = \dfrac{2}{1} = 2 \).
- x-intercept: Solve \( 2x + 3 = 0 \Rightarrow x = -\dfrac{3}{2} \).
- y-intercept: \( x = 0 \Rightarrow y = \dfrac{3}{-1} = -3 \).
