Topic 2.14 · HL only

Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions: notes and practice questions

Summary

Odd Functions: Symmetric about the origin ( $f(-x) = -f(x)$ ). Examples: $x^3$ , $\sin x$ .
Even Functions: Symmetric about the y-axis ( $f(-x) = f(x)$ ). Examples: $x^2$ , $\cos x$ .
Inverse Functions: Exist only for one-to-one functions (pass Horizontal Line Test). Domain of $f^{-1}$ is Range of $f$ , and vice-versa.
Finding Inverses: Swap $x$ and $y$ , then solve for $y$ .
Symmetry: $f(x)$ and $f^{-1}(x)$ are reflections across $y=x$ .
Restricted Domains: Necessary for non-one-to-one functions (e.g., $x^2$ ) to define inverses.
Inverse Trig: Specific domain/range restrictions apply (arcsin, arccos, arctan).
Derivative of Inverse: $(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$ .

How it is examined

Domain restriction is the examinable idea: a function that is not one to one has no inverse until you restrict it, and the question wants the restriction stated as well as the inverse found. Proving a function is odd or even means showing $f(-x)$ algebraically, not pointing at the graph. 4 to 6 marks.

Key ideas

Odd and even functions.
Finding the inverse function $f^{-1}(x)$ , including domain restriction.
Self-inverse functions.

Linking questions

Link to symmetry properties of trigonometric graphs (AHL 3.11).

Practice questions

5 questions · 1 easy · 1 medium · 3 hard

Showing 5 of 5

Question 1

EasyPaper 1 · no calculator9 marks

(a)

Determine algebraically whether the function is odd, even or neither.

$f(x) = x \sin(x)$

[3]

(b)

Determine algebraically whether the function is odd, even or neither.

$g(x) = \frac{x^3}{x^4 + 2}$

[3]

(c)

Determine algebraically whether the function is odd, even or neither.

$h(x) = x^2 + x$

[3]

Question 2

MediumPaper 2 · calculator6 marks

(a)

Consider the function $h(x) = e^{2x} - e^{-2x}$ , for $x \in \mathbb{R}$ .

Show that $h$ is an odd function.

[2]

(b)

The function $k$ is given by $k(x) = \frac{x+1}{x^2 - 2x - 8}$ , where $x \in \mathbb{R}, x \neq -2, x \neq 4$ .

Solve the inequality $h(x) \ge k(x)$ .

[4]

Question 3

HardPaper 2 · calculator6 marks

(a)

Consider the function $h(x) = \ln\left(\frac{1+x}{1-x}\right)$ , for $x \in (-1, 1)$ .

(a) Show that $h$ is an odd function.

[2]

(b)

The function $k$ is given by $k(x) = \frac{3x}{x^2 - 0.25}$ , where $x \in \mathbb{R}, x \neq \pm 0.5$ .

(b) Solve the inequality $h(x) \ge k(x)$ .

[4]

Question 4

HardPaper 1 · no calculator8 marks

(a)

Consider the function $g(x)=\frac{1-\cos(ax)}{x^2}$ , where $x \neq 0$ and $a \in \mathbb{R}^+$ .

(a) Show that $g$ is an even function.

[2]

(b)

(b) Given that $\lim_{x\to0} g(x) = 8$ , find the value of $a$ .

[6]

Question 5

HardPaper 1 · no calculator8 marks

(a)

Consider the function $g(x)=\frac{1-\cos(ax)}{e^{x^2}-1}$ , where $x \neq 0$ and $a \in \mathbb{R}^+$ .

(a) Show that $g$ is an even function.

[2]

(b)

(b) Given that $\lim_{x\to0} g(x) = 8$ , find the value of $a$ .

[6]

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What does Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions cover in IB Maths AA?

Odd Functions: Symmetric about the origin (f(-x) = -f(x)). Examples: x^3, sin x. Even Functions: Symmetric about the y-axis (f(-x) = f(x)). Examples: x^2, cos x. Inverse Functions: Exist only for one-to-one functions (pass Horizontal Line Test). Domain of f^-1 is Range of f, and vice-versa.

Is Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions SL or HL?

Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions is HL only. SL students are not examined on it.

How do I revise Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions for IB Maths AA?

Start from the core idea: odd Functions: Symmetric about the origin (f(-x) = -f(x)). Examples: x^3, sin x. In the exam: domain restriction is the examinable idea: a function that is not one to one has no inverse until you restrict it, and the question wants the restriction stated as well as the inverse found. Proving a function is odd or even means showing f(-x) algebraically, not pointing at the graph. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions?

FourtyFive has 5 Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

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Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Every Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions question, marked for you

Where marks are lost

Practise this topic as a session

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How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Every Odd & Even functions [f(-x) = f(x), f(-x)=-f(x)], harder inverse functions question, marked for you

Where marks are lost

Questions,answered.

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Questions,
answered.

Start with the IB question
bank built for you.