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Monthly Archives: July 2024

July 25, 2024 · 7:05 am

Function Composition and Domain and Range

My year 12 Specialist students have been working on function composition and the domain and range of the resulting composition. And they have been struggling a bit with why the composition doesn’t exist.

For example,

The functions $f$ and $g$ are defined by $f(x)=x^2-2$ and $g(x)=\sqrt{x}$

(a) Explain why $g(f(x))$ is not defined.

(b) By suitably restricting the domain of $f$ , obtain a function $f_1$ such that $g(f_1(x))$ is defined.

For the composite function to exist the range of the inner function (in this case $f(x)$ ) must be a subset of the domain of the outer function (in this case $g(x)$ ).

Start by finding the domain and range of each function.

We can see the range of $f(x)$ is not a subset of the domain of $g(x)$

i.e. $R_{f(x)}\nsubseteq D_{g(x)}$

We can restrict the range of $f(x)$ by restricting the domain.

$f(x)\geq0$

$x^2-2\geq0$

$x^2\geq2$

$x\leq -\sqrt{2}$ or $x\geq \sqrt{2}$

Therefore $f_1(x)=x^2-2, x\leq -\sqrt{2}$ or $x\geq \sqrt{2}$

and $g(f_1(x))=\sqrt{x^2-2}, x\leq -\sqrt{2}$ or $x\geq \sqrt{2}$

Australian Mathematics Competition – Polynomial Question

I came across this question from the 2010 Senior Australian Mathematics Competition:

A polynomial $f$ is given. All we know about it is that all its coefficients are non-negative integers, $f(1)=6$ and $f(7)=3438$ . What is the value of $f(3)$
Australian Mathematics Competition 2006-2012

I thought ‘excellent, a somewhat hard polynomial question for my students’ and then I tried it. Now I know why only 1% of students got it correct.

As we don’t know the order of the polynomial, let

$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x^1+a_0$

We know all of the coefficients are greater than or equal to zero. We also know

$f(1)=a_n+a_{n-1}+...+a+a_0=6$

Which means that all of the coefficients are between zero and six

$0\le{a_n}\le{6}$

We have also been given $f(7)$

$f(7)=7^na_n+7^{n-1}a_{n-1}+ ... +7a+1=3438$

As all of the coefficients are between zero and six, this is $3438$ written in base 7.

Let’s calculate a few powers of 7

	Powers of 7
$7^0$	1
$7^1$	7
$7^2$	49
$7^3$	343
$7^4$	2401
$7^5$	16807

Hence $3438$ written in base $7$ is $13011$

Therefore $f(x)=x^4+3x^3+x+1$

$f(3)=3^4+3\times3^3+3+1$

$f(3)=81+81+4$

$f(3)=166$

I really like this question. I think it could work well as a class extension activity with a bit of scaffolding.

As numbers	As Powers of 7
$3438=1\times2401+1037$	$3438=1\times7^4+1037$
$1037=3\times343+8$	$1037=3\times7^3+8$
$8=1\times7+1$	$8=1\times7^1+1$
$1=1\times1$	$1=1\times7^0$

Monthly Archives: July 2024

Function Composition and Domain and Range

Australian Mathematics Competition – Polynomial Question

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