IB Maths AA HL 2.14 Odd/even and inverse functions Question Bank

Practise IB Mathematics HL 2.14 by applying odd/even and inverse functions methods to exam-style questions.

Exam points

identify the mathematical structure, variable or representation
select and apply the correct theorem, formula or algorithm
check the result using units, domain, graph or logical reasoning

[Maximum number: 2]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

State whether the function $f(x)=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ is odd, even or neither. Justify your answer. [2]

Now consider the general curve given by $y=\frac{x\left(x^{2}-A\right)}{x^{2}+A}$ , where A is a positive constant
and $x \in \mathbb{R}$ . and $x \in \mathbb{R}$ .

EITHER
odd A1
since $f(-x)=\frac{(-x)\left((-x)^{2}-16\right)}{(-x)^{2}+16}=\frac{-x\left(x^{2}-16\right)}{x^{2}+16}=-f(x)$ (or equivalent) $\quad \boldsymbol{R} 1$
OR
odd A1
since graph of f has rotational symmetry order 2 (of 180 degrees) about $(0,0) \quad \boldsymbol{R}$ 1

OR
odd A1 since graph has reflection symmetry over the x-axis followed by reflection symmetry over the y-axis (or vice versa)

R1

Note: Do not award A1R0.

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AHL 2.14 (HL)—Odd/even and inverse functions question 1