SL 2.8—Reciprocal and simple rational functions, equations of asymptotes Questionbank

Practice SL 2.8—Reciprocal and simple rational functions, equations of asymptotes 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

SLPaper 1

Let $g(x)=\frac{3 x-6}{x+2}$ , for $x \neq-2$ .

For the graph of $g$ , find the $y$ -intercept.

[2]

Hence or otherwise, write down $\lim _{x \rightarrow \infty}\left(\frac{3 x-6}{x+2}\right)$ .

[2]

Question 2

SLPaper 1

The function $g$ is defined by $g(x)=\frac{4}{x+1}$ , where $x \in \mathbb{R}, x \neq-1$ .

Write down the equation of the vertical asymptote of the graph of $g$ .

[1]

Write down the equation of the horizontal asymptote of the graph of $g$ .

[1]

(i) Find the coordinates where the graph of $g$ crosses the $y$ -axis.

[1]

Find the coordinates where the graph of $g$ crosses the $x$ -axis.

[1]

Sketch the graph of $g$ on the axes below.

[1]

Question 3

SLPaper 1

The function $f$ is defined by $f(x)=\frac{3}{x-2}$ , where $x \in \mathbb{R}, x \neq 2$ .

Write down the equation of the vertical asymptote of the graph of $f$ .

[1]

Write down the equation of the horizontal asymptote of the graph of $f$ .

[1]

Find the coordinates where the graph of $f$ crosses the $y$ -axis.

[1]

Find the coordinates where the graph of $f$ crosses the $x$ -axis.

[1]

Sketch the graph of $f$ on the axes below.

[1]

Question 4

SLPaper 1

Let $f(x)=\frac{4 x+2}{2 x-1}$ , for $x \neq \frac{1}{2}$ .

For the graph of $f$ , find the $y$ -intercept.

[2]

Hence or otherwise, write down $\lim _{x \rightarrow \infty}\left(\frac{4 x+2}{2 x-1}\right)$ .

[2]

Question 5

SLPaper 2

A rational function has vertical asymptote $x=4$ and horizontal asymptote $y=1$ . It passes through $(0,-1)$ and $(5,9)$ .

Show $f(x)=1+\dfrac{k}{x-4}$ and determine $k$ . Hence express $f(x)=\dfrac{ax+b}{cx+d}$ with integers.

[4]

Determine domain, range, intercepts, asymptotes, and the sign of $f(x)-1$ on $(- \infty,4)$ , $(4,\infty)$ .

[5]

Solve $f(x)=g(x)$ to 3 d.p. and justify one solution in each of $(-1,4)$ and $(4,\infty)$ .

[6]

Describe the transformations from $y=1/x$ to $y=f(x)$ .

[2]

Question 6

SLPaper 2

A rational function has vertical asymptote $x=4$ and horizontal asymptote $y=7$ . It passes through the points $(0,5)$ and $(6,11)$ .

Show that the function can be written in the form $f(x)=7+\frac{k}{x-4}$ for some constant $k$ , and determine $k$ . Hence write $f(x)$ explicitly as $\dfrac{ax+b}{cx+d}$ with integer coefficients.

[4]

Determine all key features of $f$ : domain, range, intercepts, asymptotes, and the sign of $f(x)-7$ on $(- \infty,4)$ and $(4,\infty)$ .

[5]

Solve $f(x)=g(x)$ to three decimal places and justify exactly one solution in each of $(-1,4)$ and $(4,\infty)$ .

[6]

Describe transformations mapping $y=1/x$ to $y=f(x)$ .

[2]

Question 7

SLPaper 2

A rational function has vertical asymptote $x=1$ and horizontal asymptote $y=5$ . It passes through $(0,2)$ and $(3,\tfrac{13}{2})$ .

Show $f(x)=5+\dfrac{k}{x-1}$ and find $k$ . Hence write $f(x)=\dfrac{ax+b}{cx+d}$ with integers.

[4]

Determine domain, range, intercepts, asymptotes, and the sign of $f(x)-5$ on $(- \infty,1)$ and $(1,\infty)$ .

[5]

Solve $f(x)=g(x)$ to 3 d.p. and justify one solution in each of $(-2,1)$ and $(1,\infty)$ .

[6]

Describe transformations from $y=1/x$ to $y=f(x)$ .

[2]

Question 8

SLPaper 2

A rational function has vertical asymptote $x=-4$ and horizontal asymptote $y=2$ . It passes through the points $(0,3)$ and $(2,\tfrac{8}{3})$ .

Show that the function can be written in the form $f(x)=2+\dfrac{k}{x+4}$ ; determine $k$ . Hence write $f(x)=\dfrac{ax+b}{cx+d}$ with integer coefficients.

[4]

Determine domain, range, intercepts, asymptotes, and the sign of $f(x)-2$ on $(- \infty,-4)$ and $(-4,\infty)$ .

[5]

Solve $f(x)=g(x)$ to three decimal places and justify uniqueness of one solution in each of $(-5,-4)$ and $(-4,\infty)$ .

[6]

Describe transformations mapping $y=1/x$ to $y=f(x)$ .

[2]

Question 9

SLPaper 2

Consider the function $g(x)=\frac{p x+1}{q x-r}, x \neq \frac{r}{q}$ , where $p, q, r \in \mathbb{Z}$ . The graph of $y=g(x)$ has a vertical asymptote at $x=-1$ and a horizontal asymptote at $y=2$ . It intersects the $y$ -axis at the point $\mathrm{C}\left(0,-\frac{1}{3}\right)$ .

Write down the equations of the asymptotes of the graph of $y=g(x)$ .

[2]

Determine the values of $p, q$ , and $r$ .

[3]

Find the $x$ -intercept of the graph of $y=g(x)$ .

[2]

Sketch the graph of $y=g(x)$ , indicating the asymptotes and the intercepts.

[3]

Question 10

SLPaper 2

Consider the function $f(x)=\frac{3 x+6}{x-4}$ , where $x \in \mathbb{R}, x \neq 4$ .

Find the equations of the vertical and horizontal asymptotes of the graph of $f$ . [

[2]

Determine the coordinates of the points where the graph of $f$ intersects the $x$ -axis and $y$ -axis.

[2]

Find the limit as $x$ approaches negative infinity, i.e., $\lim _{x \rightarrow-\infty} f(x)$ .

[2]

Sketch the graph of $f$ , clearly indicating the asymptotes and axis intercepts.

[3]