IB Mathematics Analysis and Approaches Topic AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction and mirrors Paper 1, 2, 3 style where relevant.

Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.

Paper

Level

Question 1

HLPaper 3

This question investigates iterated functions and their long-term behaviour.

Given a function $f$ , the iteration sequence starting at $x_{0}$ is defined by

$x_{n + 1} = f (x_{n}), n = 0, 1, 2, \dots$

A value $L$ such that $f (L) = L$ is called a fixed point of $f$ .

Consider the function $f (x) = cos x$ , where $x$ is in radians.

Starting with $x_{0} = 1$ , find the values of $x_{1}$ , $x_{2}$ and $x_{3}$ , giving your answers correct to three decimal places.

[3]

Suppose the iteration sequence converges to a limit $α$ . Explain why $α$ must satisfy $cos α = α$ .

[2]

Use your graphic display calculator to find the value of $α$ , correct to three decimal places.

[2]

The behaviour of such iteration sequences can be visualised using a cobweb diagram. On the same axes, one draws $y = f (x)$ and $y = x$ . Starting from $(x_{0}, 0)$ , vertical and horizontal lines are drawn alternately between the two graphs, tracing the path of the iteration.

Now consider the family of functions $g (x) = k x (1 - x)$ , where $k$ is a positive real constant.

Find, in terms of $k$ , the two fixed points of $g$ .

[3]

For $k = 2.5$ and $x_{0} = 0.1$ , calculate $x_{1}$ , $x_{2}$ and $x_{3}$ , and comment on the behaviour of the sequence in relation to the non-zero fixed point.

[3]

To understand why some fixed points attract nearby iterates, we study the behaviour of the error $e_{n} = x_{n} - L$ , where $L$ is a fixed point.

For $x_{n}$ close to $L$ , we may use the approximation $f (x) \approx f (L) + f^{'} (L) (x - L)$ .

Using this approximation, show that $e_{n + 1} \approx f^{'} (L) \cdot e_{n}$ .

[3]

Hence explain why the fixed point $L$ attracts nearby iterates when $∣ f^{'} (L) ∣ < 1$ , and repels them when $∣ f^{'} (L) ∣ > 1$ .

[3]

Show that $g^{'} (x^{*}) = 2 - k$ , where $x^{*} = \frac{k - 1}{k}$ is the non-zero fixed point of $g$ .

[3]

Hence determine the range of values of $k$ for which the non-zero fixed point $x^{*}$ is an attracting fixed point in the interval $(0, 1)$ .

[3]

10.

Verify that your result from part 7 is consistent with the convergence observed in part 3 for $f (x) = cos x$ .

[2]

11.

For $k = 3.2$ , the non-zero fixed point is no longer attracting. Starting with $x_{0} = 0.2$ , use your GDC to iterate the function $g (x) = 3.2, x (1 - x)$ and describe the long-term behaviour of the sequence. State the values the sequence approaches, correct to three decimal places.

[2]

Question 2

HLPaper 2

Let $f (x) = x^{3} - 6 x^{2} + 9 x + 1$ , for $x \in R$ .

Find $f^{'} (x)$ and hence find the coordinates of the stationary points of $f$ .

[5]

Use $f^{''} (x)$ to classify the stationary points.

[3]

Sketch the graph of $y = f (x)$ , labelling the stationary points, the $y$ -intercept and the $x$ -intercept (giving values to three significant figures where necessary).

[4]

Find the $x$ -coordinate of the point of inflexion of the graph of $y = f (x)$ .

[2]

Justify why $g^{- 1}$ exists.

[1]

State the domain and range of $g^{- 1}$ .

[2]

Find the value of $g^{- 1} (2.5)$ , giving your answer to three significant figures.

[2]

Question 3

HLPaper 1

Let $f (x) = \frac{2 x + 3}{3 x - 4}$ , for $x \neq = \frac{4}{3}$ .

The following diagram shows part of the graph of $y = f (x)$ .

For the graph of $y = f (x)$ , find the coordinates of:

the $x$ -intercept;

the $y$ -intercept.

[3]

For the graph of $y = f (x)$ , find the equation of:

(i) the horizontal asymptote;

(ii) the vertical asymptote.

[3]

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

[3]

The graphs of $f (x)$ and $f^{- 1} (x)$ have two points of intersection.

Find the coordinates of the two points of intersection. Give your answers in exact form.

[4]

Hence find the equation of the perpendicular bisector of the segment joining the two points of intersection.

[4]

Question 4

HLPaper 2

Define $f(x)=\frac{x-1}{kx+2},\quad k\in\mathbb R,\ x\ne -\frac{2}{k}\ (k\ne0).$

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

[3]

Determine for which $k$ values $f$ is self-inverse.

[2]

Let $g_{k} (x) = \frac{k x + 1}{x - k}$ . Prove self-inverse and find fixed points.

[3]

For $k > 0$ , solve $g_{k} (x) \geq x$ . For $k = 2$ , solve $g_{k} (x) = ln (1 + x)$ to 3 s.f., noting the number of roots.

[6]

Question 5

HLPaper 1

Given $f(x)=\frac{mx+n}{px-q},\quad mq+np\ne0,\quad x\ne \frac{q}{p}.$

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

[3]

Show that $f$ is a self-inverse function if and only if $m - q = 0$ .

[4]

Let $g_{s} (x) = \frac{s x + 1}{x - s}$ where $s \neq = 0$ . Prove that $g_{s}$ is a self-inverse function and find its fixed points.

[3]

For $s > 0$ , solve the inequality $g_{s} (x) \geq x$ .

[4]

IB Mathematics Analysis and Approaches AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank

Unlock all 21 questions with a free account

IB Mathematics Analysis and Approaches AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank

Unlock all 21 questions with a free account