Two-way Functions

Age 16 to 18

Well done to Pablo and Sergio from Kings College of Alicante and Jay from St Stephens Carramar who both found ways to complete the two-way function table.

Here is Pablo's and Sergio's solution.

	$y$ -axis is an asymptote	Symmetrical about $x=1$	Intercepts the y-axis at 3	passes through origin
$x = 1$ is a root	$y=\ln{x}$	$y = \vert x-1\vert$	$y = -3x+3$	$y=x^2(x-1)$
has exactly two roots	$y=\frac{1}{x^2}-3$ for $x \neq 0$	$y = x(x-2)$	$y=-x^2+2x+3$	$y=x(x-5)^2$
x-axis is an asymptote	$y = \dfrac{1}{x}$ for $x \neq 0$	$y = \dfrac{1}{(x-1)^2}$ for $x \neq 1$	$y = \dfrac{3}{x+1}$ for $x \neq -1$	$y=\frac{x}{(x-1)^2}$ for $x \neq 1$
$y \to \infty$ as $x \to \infty$	$y=\frac{x^2+1}{x}$ for $x \neq 0$	$y=(x-1)^2$	$y = 2+(x-1)^4$	$y=x$

Here is Jay's solution.

	$y$ -axis is an asymptote	Line of symmetry is $x = 1$	Y intercept = (0,3)	passes through origin
$x = 1$ is a root	$y = \frac{1}{x}-1$	$y = \vert x-1\vert$	$y = -3x+3$	$y=(2x-1)^2-1$
has exactly two roots	$y = (\frac{1}{x}-2)(\frac{1}{x}+2)$	$y = x(x-2)$	$y=(x-3)^2-6$	$y=(2x-1)^2-1$
x-axis is an asymptote	$y = \dfrac{1}{x}$ for $x \neq 0$	$y = \dfrac{1}{(x-1)^2}$ for $x \neq 1$	$y = \dfrac{3}{x+1}$ for $x \neq -1$	$y=(\frac{1}{x+1}-1)(\frac{1}{x-1})$
$y \to \infty$ as $x \to \infty$	$y = \frac{2^x}{x}$	$y = (x-1)^2$	$y = 2+(x-1)^4$	$y=x^2$

This comes in two parts, with the first being less fiendish than the second. Itâ€™s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.

Discriminating

You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

Factorisable Quadratics

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners

Two-way Functions

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