In VCE Specialist Mathematics, a rational function is defined as the ratio of two polynomials. These functions exhibit unique asymptotic behavior and specific domain restrictions that are central to the study of functions and graphs.
A rational function \(f\) has a rule of the form:
\$\(f(x) = \frac{P(x)}{Q(x)}\)\$
where \(P(x)\) and \(Q(x)\) are polynomials.
For sketching and identifying asymptotes, it is often necessary to perform polynomial long division to express the function in the form:
\$\(f(x) = q(x) + \frac{r(x)}{Q(x)}\)\$
Where:
* \(q(x)\) is the quotient (representing the non-vertical asymptote).
* \(r(x)\) is the remainder, where the degree of \(r(x) <\) degree of \(Q(x)\).
KEY TAKEAWAY: Converting a rational function into quotient-remainder form is the most effective first step for identifying the behavior of the graph as \(x \to \pm\infty\).
The domain of a rational function \(f(x) = \frac{P(x)}{Q(x)}\) is all real numbers except where the denominator is zero:
\$\(\text{Domain} = \{x \in \mathbb{R} : Q(x) \neq 0\}\)\$
Points excluded from the domain usually correspond to vertical asymptotes.
COMMON MISTAKE: Students often forget to check if the \(x\)-intercepts found by setting \(P(x)=0\) are actually in the domain. If a value makes both \(P(x)=0\) and \(Q(x)=0\), there may be a “hole” (point of discontinuity) rather than an intercept or asymptote.
Asymptotes are lines that the graph approaches as \(x\) or \(y\) tends toward infinity.
Vertical asymptotes occur at the values of \(x\) that make the denominator \(Q(x) = 0\) (assuming no common factors with the numerator).
* If \((x - a)\) is a factor of \(Q(x)\) but not \(P(x)\), then \(x = a\) is a vertical asymptote.
* As \(x \to a^+\), \(f(x) \to \pm\infty\).
* As \(x \to a^-\), \(f(x) \to \pm\infty\).
The behavior of the graph as \(x \to \pm\infty\) is determined by the quotient \(q(x)\) after division.
| Degree Relationship | Type of Asymptote | Equation |
|---|---|---|
| \(\text{deg}(P) < \text{deg}(Q)\) | Horizontal | \(y = 0\) (the \(x\)-axis) |
| \(\text{deg}(P) = \text{deg}(Q)\) | Horizontal | \(y = \frac{a_n}{b_n}\) (ratio of leading coefficients) |
| \(\text{deg}(P) = \text{deg}(Q) + 1\) | Oblique (Slant) | \(y = mx + c\) (the linear quotient) |
| \(\text{deg}(P) > \text{deg}(Q) + 1\) | Parabolic/Higher order | \(y = q(x)\) (the polynomial quotient) |
To determine if a graph approaches an asymptote from above or below:
1. Consider \(f(x) = q(x) + \frac{r(x)}{Q(x)}\).
2. As \(x \to \pm\infty\), the term \(\frac{r(x)}{Q(x)} \to 0\).
3. If \(\frac{r(x)}{Q(x)} > 0\) as \(x \to \infty\), the graph approaches the asymptote from above.
4. If \(\frac{r(x)}{Q(x)} < 0\) as \(x \to \infty\), the graph approaches the asymptote from below.
EXAM TIP: When asked to “sketch and label all asymptotes with their equations,” ensure you use dashed lines for the asymptotes and write the full equation (e.g., \(y = x - 3\), not just \(x - 3\)).
If \(f(x) = g(x) + h(x)\), the graph of \(f(x)\) can be sketched by adding the \(y\)-values of \(g(x)\) and \(h(x)\) for each \(x\).
* Commonly used when \(f(x) = x + \frac{1}{x}\).
* The graph will approach the line \(y = x\) as \(\frac{1}{x} \to 0\).
To sketch \(y = \frac{1}{f(x)}\) from a known graph of \(y = f(x)\):
* Zeros of \(f(x)\) become vertical asymptotes of \(\frac{1}{f(x)}\).
* Local maximums of \(f(x)\) become local minimums of \(\frac{1}{f(x)}\) (and vice versa).
* Where \(f(x)\) is increasing, \(\frac{1}{f(x)}\) is decreasing.
* The points where \(f(x) = 1\) or \(f(x) = -1\) are invariant (they stay the same).
VCAA FOCUS: Specialist Mathematics exams frequently require students to find the range of a rational function. This often involves finding the coordinates of stationary points using differentiation (\(f'(x) = 0\)) and observing the behavior near asymptotes.
| Feature | Analysis Method |
|---|---|
| Vertical Asymptotes | Solve Denominator \(= 0\). |
| Non-Vertical Asymptotes | Use polynomial division; identified by the quotient \(q(x)\). |
| \(x\)-intercepts | Solve Numerator \(= 0\). |
| \(y\)-intercept | Calculate \(f(0)\). |
| Stationary Points | Solve \(f'(x) = 0\) using the Quotient Rule. |
| Range | Analyze the \(y\)-values between stationary points and asymptotes. |
STUDY HINT: Always check for symmetry. If \(f(-x) = f(x)\), the function is even (symmetric about the \(y\)-axis). If \(f(-x) = -f(x)\), the function is odd (symmetric about the origin, rotational symmetry of \(180^\circ\)). This can significantly simplify sketching.
