IB Maths AA HL · Unit 2: Functions
IB Maths AA HL Odd, Even, Inverse, Inequalities and Modulus Questions
Exam-style IB Maths AA HL odd, even, inverse, inequalities and modulus questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 33 questions
- Paper 1: 28
- Paper 2: 5
- 8 easy
- 11 medium
- 8 hard
- 6 starter
- 3 worked examples
Practise Odd, Even, Inverse, Inequalities and Modulus questions →
AA HL formula booklet
What's examined in AA HL odd, even, inverse, inequalities and modulus
The question bank covers these odd, even, inverse, inequalities and modulus question types (number of questions in brackets):
- Function Types and Inverses (15)
- Algebraic Equations & Inequalities (13)
- Graphical Transformations (5)
Key formulas
- Modulus & argument
- \(r = |z| = \sqrt{a^2 + b^2},\ \theta = \arg z\)
In the same notation as the IB formula booklet. All AA HL formulas →
Odd, Even, Inverse, Inequalities and Modulus worked examples
Worked example 1: Identifying odd/even functions algebraically · easy
Determine algebraically whether the function $f(x) = \frac{x^3}{x^2 + 1}$ is odd, even, or neither.
1. Evaluate $f(-x)$ to test the symmetry of the function.
2. Substitute $-x$ into the expression: $f(-x) = \frac{(-x)^3}{(-x)^2 + 1}$.
3. Simplify the numerator and denominator: $(-x)^3 = -x^3$ and $(-x)^2 = x^2$.
4. Rewrite the fraction: $f(-x) = \frac{-x^3}{x^2 + 1}$.
5. Factor out the negative sign: $f(-x) = -\left(\frac{x^3}{x^2 + 1}\right)$.
6. Conclude by comparing to the original function: Since $f(-x) = -f(x)$, the function is odd.
Examiner tip: Students often incorrectly classify functions as "neither" because they make sign errors when expanding negative variables raised to powers, especially $(-x)^2$. Remember that an even power always eliminates the negative.
Worked example 2: Inverses with restricted domains · medium
The function $f(x) = (x-3)^2 - 1$ is defined for the restricted domain $x \ge 3$. Find an expression for its inverse, $f^{-1}(x)$.
1. Set the function equal to $y$: $y = (x-3)^2 - 1$.
2. Swap the $x$ and $y$ variables to find the inverse mapping: $x = (y-3)^2 - 1$.
3. Rearrange to isolate the squared term: $x + 1 = (y-3)^2$.
4. Square root both sides, remembering the $\pm$ symbol: $y - 3 = \pm\sqrt{x+1}$.
5. Select the correct root. Since the original domain was $x \ge 3$, the range of the inverse must be $y \ge 3$. Thus, we must take the positive root: $y - 3 = +\sqrt{x+1}$.
6. State the final inverse function: $\mathbf{f^{-1}(x) = \sqrt{x+1} + 3}$.
Examiner tip: Forgetting the $\pm$ when taking the square root, and therefore failing to explicitly justify why the positive root is chosen based on the domain restriction, will cost you reasoning marks in Paper 1.
Worked example 3: Solving modulus inequalities · hard
Solve the inequality $|2x - 5| \ge |x + 1|$ algebraically.
1. Square both sides of the inequality. Since both sides are absolute values and strictly non-negative, squaring preserves the inequality and removes the modulus: $(2x - 5)^2 \ge (x + 1)^2$.
2. Expand both binomials: $4x^2 - 20x + 25 \ge x^2 + 2x + 1$.
3. Rearrange all terms to one side to form a quadratic inequality: $3x^2 - 22x + 24 \ge 0$.
4. Factorise the quadratic to find the critical values: $(3x - 4)(x - 6) \ge 0$.
5. Identify the critical values where the expression equals zero: $x = \frac{4}{3}$ and $x = 6$.
6. Determine the valid regions. Since it is a "greater than or equal to" ($\ge$) inequality for a positive parabola, the solution lies outside the roots: $\mathbf{x \le \frac{4}{3}}$ or $\mathbf{x \ge 6}$.
Examiner tip: Squaring both sides is the most efficient algebraic method when there is a modulus on both sides. Trying to split this into multiple piecewise cases often leads to messy intersecting domains and lost solutions.
FAQ
How many IB Maths AA HL odd, even, inverse, inequalities and modulus questions are there?
There are 33 exam-style odd, even, inverse, inequalities and modulus questions in the AA HL question bank (Paper 1: 28 · Paper 2: 5), graded 8 easy, 11 medium, 8 hard, 6 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is odd, even, inverse, inequalities and modulus on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 28 · Paper 2: 5. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA HL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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