AHL 2.13—Rational functions Questionbank

Practice AHL 2.13—Rational functions with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider the function $f(x)=\frac{3-x}{x+1}$ , where $x \in \mathbb{R}, x \neq-1$ .

Find the coordinates of the points where the graph of $y=f(x)$ intersects the axes.

[3]

Sketch the graph of $y=f(x)$ , clearly showing the asymptotes and the points found in part 1.

[4]

Question 2

HLPaper 1

Consider the function $f(x)=\frac{2 x+1}{x-3}$ , where $x \in \mathbb{R}, x \neq 3$ .

Find the equations of all asymptotes of $f$ .

[2]

Find the coordinates of the point where the graph of $y=f(x)$ intersects the $y$ -axis.

[2]

Question 3

HLPaper 1

Consider the function $f(x)=\frac{x^{2}+2 x-8}{2 x-5}$ , where $x \in \mathbb{R}, x \neq \frac{5}{2}$ .

Find the coordinates of all intercepts of the graph of $y=f(x)$ with the $x$ - and $y$ -axes.

[3]

Find the equation of the oblique asymptote of $f$ .

[3]

Express $\frac{1}{f(x)}$ in partial fractions.

[3]

Hence, find the exact value of $\int_{3}^{4} \frac{1}{f(x)} d x$ , expressing your answer as a single logarithm.

[3]

Question 4

HLPaper 1

Consider the function $f(x)=\frac{x^{2}-2 x-3}{x+2}$ , where $x \in \mathbb{R}, x \neq-2$ .

Find the coordinates of all intercepts of the graph of $y=f(x)$ with the axes.

[3]

Find the equations of all asymptotes of $f$ .

[3]

Find the $x$ -coordinates of the stationary points of the graph of $f$ .

[3]

Question 5

HLPaper 1

Let $f(x)=\frac{x^{2}+px+q}{x-3},\, x\ne3.$

Show that $f(x)=x+p+3+\dfrac{q+3p+9}{x-3}$ and state the equation of the oblique asymptote.

[3]

Prove that $f$ is neither even nor odd for any $p,q$ .

[4]

Find the conditions on $p$ and $q$ such that the graph of $f$ has exactly one $x$ -intercept, and give the coordinate of this intercept in terms of $p$ .

[7]

For $q+3p+9\ne0$ and $\Delta>0$ , solve the inequality $f(x)\ge x+p+3$ .

[6]

Question 6

HLPaper 2

Consider $f(x)=\frac{x^3+4x^2+x+\lambda}{x^2 - x - 6},\qquad x\in\mathbb{R}\setminus\{-2,3\},\ \lambda\in\mathbb{R}\setminus\{-6, -66\}.$

Show the vertical asymptotes are independent of $\lambda$ and find them.

[2]

Write $f(x)=q(x)+\dfrac{r(x)}{x^2-x-6}$ with $q$ linear and $\deg r\le1$ .

[4]

Deduce the oblique asymptote.

[2]

Find the intersection point(s) of the graph with its oblique asymptote.

[4]

Solve $f(x)\ge x+5$ for $\lambda=-30$ .

[6]

Question 7

HLPaper 1

Let

f(x)=\frac{x^{2}+px+q}{x+3},\,x\ne-3.

Divide to obtain $f(x)=Q(x)+\dfrac{R}{x+3}$ and state the equation of the slant asymptote.

[3]

Show that $f$ is neither even nor odd for any $p,q$ .

[2]

Find the conditions on $p$ and $q$ for the graph of $f$ to have exactly one $x$ -intercept, and state the coordinates of the intercept in each case.

[7]

For $q-3p+9\ne0$ and $\Delta>0$ , solve the inequality $f(x)\ge x+p-3$ .

[5]

Question 8

HLPaper 2

The function $f$ is defined by $f(x)=\frac{x^3+2x^2+6x+\lambda}{x^2-2x-3}$ for $x \in \mathbb{R}, x \neq -1, 3$ , where $\lambda \in \mathbb{R}$ and $\lambda \neq 5, -63$ .

Find the equations of the vertical asymptotes and justify why they are independent of $\lambda$ .

[2]

Express $f(x)$ in the form $f(x)=q(x)+\dfrac{r(x)}{x^2-2x-3}$ , where $q(x)$ is a linear function.

[4]

State the equation of the slant asymptote.

[2]

Find the coordinates of the point where the graph of $f$ intersects its slant asymptote.

[4]

Given that $\lambda=-12$ , solve the inequality $f(x) \ge x+4$ .

[6]

Question 9

HLPaper 2

Given $f(x)=\frac{ax+2}{-x+a},\quad a\in\mathbb{R},\ x\ne a.$

Find $f^{-1}(x)$ and its domain.

[3]

Determine when $f$ is self-inverse.

[2]

For $a=0$ , find fixed points and solve $f(x)\ge x$ .

[4]

For $a=1$ , solve $f(x)=\sqrt{x+5}$ numerically to 3 s.f., stating the number of solutions.

[3]

Question 10

HLPaper 1

Consider the function $f(x)=\frac{x^{2}+2}{x-1}$ , where $x \in \mathbb{R}, x \neq 1$ .

Find the coordinates of all stationary points of $f$ .

[5]

Sketch the graph of $y=f(x)$ , clearly showing the asymptotes, intercepts, and stationary points.

[4]