IB Maths AA HL Ahl 2 14 Hl Odd Even and Inverse Functions Questions

Practise testing odd or even functions, deciding when inverses exist, constructing inverse functions and checking domain, range and reflection conditions.

Exam points

Classify functions as odd or even from algebraic and graph symmetry.
Determine and apply inverse functions, restrictions and symmetry using complete evidence.

[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.

All question bank results loaded

IB Maths AA HL Ahl 2 14 Hl Odd Even and Inverse Functions Questions question 1