IB Mathematics AA SL The rational function Study Notes
IB Mathematics AA SL The rational function Study Notes Offer a clear explanation of The rational function , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic The rational function
Reciprocal Function \( f(x) = \frac{1}{x} \)
Reciprocal Function \( f(x) = \frac{1}{x} \)
The reciprocal function is defined as:
\( f(x) = \frac{1}{x}, \quad x \neq 0 \)
Domain: \( x \in \mathbb{R}, x \neq 0 \)
Range: \( y \in \mathbb{R}, y \neq 0 \)
Asymptotes:
Vertical asymptote: \( x = 0 \)
Horizontal asymptote: \( y = 0 \)
Symmetry: The graph is symmetric about the origin (odd function)
Self-inverse nature: The reciprocal function is its own inverse:
\( f^{-1}(x) = f(x) \)
Its graph is unchanged when reflected in the line \( y = x \).
Example: Graph of \( f(x) = \frac{1}{x} \)
▶️Answer/Explanation
The graph consists of two branches:
For \( x > 0 \), the graph is in the first quadrant, approaching both axes but never touching them.
For \( x < 0 \), the graph is in the third quadrant, with similar behavior.
The graph is symmetric about the line \( y = x \) (because it is self-inverse).
Key points: \( (1,1) \) and \( (-1,-1) \)
Rational Function \( f(x) = \frac{ax + b}{cx + d} \)
Rational Function \( f(x) = \frac{ax + b}{cx + d} \)
This is a rational function where both numerator and denominator are linear expressions.
Domain: All real numbers except \( x = -\frac{d}{c} \), where the denominator is zero.
Vertical asymptote: \( x = -\frac{d}{c} \)
Horizontal asymptote: \( y = \frac{a}{c} \) (as \( x \to \pm \infty \))
Intercepts:
y-intercept: \( f(0) = \frac{b}{d} \) (if \( d \neq 0 \))
x-intercept: Solve \( ax + b = 0 \Rightarrow x = -\frac{b}{a} \) (if \( a \neq 0 \))
Graph shape: Typically resembles a hyperbola shifted and scaled according to coefficients.
Example: Sketch the graph of \( f(x) = \frac{2x + 1}{x – 3} \)
▶️Answer/Explanation
Vertical asymptote: \( x = 3 \)
Horizontal asymptote: \( y = 2 \)
x-intercept: Set numerator zero: \( 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \)
y-intercept: \( f(0) = \frac{1}{-3} = -\frac{1}{3} \)
Graph :
indicates vertical asymptote at \( x = 3 \). The graph approaches \( y = 2 \) as \( x \to \pm \infty \).
Vertical and Horizontal Asymptotes
Vertical Asymptotes: Occur where the denominator of the rational function is zero (and the numerator is not zero at the same point).
For a rational function \( f(x) = \frac{ax + b}{cx + d} \):
- The vertical asymptote is at \( x = -\frac{d}{c} \), provided \( c \neq 0 \).
Horizontal Asymptotes: Describe the behavior of the function as \( x \to \pm \infty \).
For \( f(x) = \frac{ax + b}{cx + d} \):
- If degrees of numerator and denominator are equal → horizontal asymptote is \( y = \frac{a}{c} \).
- If degree of numerator < denominator → horizontal asymptote is \( y = 0 \).
- If degree of numerator > denominator → no horizontal asymptote (possible oblique asymptote).
Example: Find the vertical and horizontal asymptotes of \( f(x) = \frac{3x – 5}{2x + 4} \).
▶️Answer/Explanation
Vertical asymptote: Set denominator zero: \( 2x + 4 = 0 \Rightarrow x = -2 \).
Horizontal asymptote: Coefficients of highest degree terms: \( y = \frac{3}{2} \).
