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AHL 2.13—Rational functions - IB Questionbank | RevisionDojo

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 (a).

[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 $f$ .

[4]

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 $\int_{2}^{4} \frac{1}{f(x)} d x$ , 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]

Show that $f$ has no stationary points.

[2]

Question 5

HLPaper 2

Let $f(x)=\frac{2 x}{x-1}$ and $g(x)=\frac{x+1}{x-2}$ , where $x \in \mathbb{R}, x \neq 1$ for $f$ , and $x \neq 2$ for $g$ .

Show that $f$ is an odd function.

[2]

Solve the inequality $f(x) \leq g(x)$ .

[4]

Question 6

HLPaper 2

Consider the rational function $f(x)=\frac{6 x^{2}-5 x-4}{2 x^{2}-8}$ , where $x \in \mathbb{R}, x \neq \pm 2$ . The graph of $y=f(x)$ has a local minimum at point P and a local maximum at point Q .

Express $f(x)$ in the form $\frac{a(x-b)(x-c)}{(x-d)(x-e)}$ , where $a, b, c, d, e \in \mathbb{R}$ .

[3]

Determine the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$ .

[2]

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

[2]

Using a graphical calculator, find the coordinates of the local minimum at P and the local maximum at Q , and hence state the range of $f(x)$ .

[3]

Solve the inequality $\frac{6|x|^{2}-5|x|-4}{2|x|^{2}-8} \geq \frac{3}{2}$ , considering the domain of the function.

[4]

Question 7

HLPaper 1

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

Express $f(x)=x+p+3+\dfrac{q+3p+9}{x-3}$ and state the slant asymptote.

[3]

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

[4]

Parameters for exactly one $x$ -intercept; give its coordinate.

[7]

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

[6]

Question 8

HLPaper 2

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

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 9

HLPaper 1

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

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

[3]

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

[4]

Find all $(p,q)$ for exactly one $x$ -intercept; give it.

[7]

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

[6]

Question 10

HLPaper 2

Define $f(x)=\frac{x^3+2x^2+6x+\lambda}{x^2-2x-3},\qquad x\neq-1,3.$

Find the vertical asymptotes and justify independence from $\lambda$ .

[2]

Express $f(x)=q(x)+\dfrac{r(x)}{x^2-2x-3}$ with $q$ linear.

[4]

State the slant asymptote.

[2]

Find the intersection with $y=x+4$ .

[4]

Solve $f(x)\ge x+4$ for $\lambda=-12$ .

[6]

AHL 2.13—Rational functions Questionbank