Problem solving with functions and proof | Maths | Quizalize & Oak National Academy

Starter quiz

${\text{fg}}(x)$ , ${\text{gh}}(x)$ , and ${\text{fgh}}(x)$ are all examples of __________.

equations

quadratic equations

multiple functions

composite functions ✓
If ${\text{f}}(x)=5x−1$ and ${\text{g}}(x)=2x+8$ which is an expression for ${\text{fg}}(x)$ ?

$7x+7$

$10x+40$

$10x+39$ ✓

$10x+8$

$10x+6$
If ${\text{f}}(x)=5x−1$ then $7{\text{f}}(8)=$ ______.

'273' ✓
If ${\text{f}}(x)=x+5$ and ${\text{g}}(x)=x^2+4x-7$ which is an expression for ${\text{gf}}(x)$ ?

$5(x+5)-7$

$(x+5)^2+4x-7$

$x^2+14x+38$ ✓

$x^2+4x+3$

$x^2+10x+25$
$x^2=100$ has how many solutions?

$0$

$1$

$2$ ✓

$3$
Which of the below are accurate manipulations of the equation $2x=y+3$ ?

$3=y-2x$

$2xy=y^2+3y$ ✓

$2xy=y^2+3$

$2x(y-3)=(y+3)(y-3)$ ✓

$2xy-6x=y^2-9$ ✓

If ${\text{f}}(x)=5x$ and ${\text{g}}(x)=7-x$ then $5(7-x)$ is __________ for ${\text{fg}}(x)$ .

an expression ✓

an equation

a formula
If ${\text{f}}(x)=5x-18$ which is an expression for ${\text{f}}(x+3)$ ?

$5(x+3)$

$5x-15$

$5x-3$ ✓

$5x+3$
If ${\text{f}}(x)=5x-18$ and ${\text{g}}(x)=-2x$ which is an expression for $2{\text{gf}}(5x)$ ?

$72-20x$

$72-100x$ ✓

$100x+72$

$100x-72$

$-2(5x-18)$
${\text{f}}(x)=k(x-3)$ If ${\text{f}}(-1)=-24$ find $k$ where $k$ is a constant.

$k=-3$

$k=1$

$k=1$

$k=3$

$k=6$ ✓
Where is the error in this proof?

$a^2-b^2=ab-b^2$ as you can’t subtract a square from both sides.

$(a+b)(a-b)=b(a-b)$ because $a-b=0$ and you can’t multiply by zero.

From $(a+b)(a-b)=b(a-b)$ to $a+b=b$ because dividing by zero is undefined. ✓

$a^2=a\times{a}$ , not $a^2=a\times{b}$
Where is the error in this proof?

$(c+a)(c-a)$ does not expand to $c^2-a^2$

$c^2-2bc=a^2-2ab$ is not an accurate rearrangement of $c^2-a^2=2bc-2ab$

$c^2-2bc+b^2$ does not factorise to $(c-b)^2$

From $(c-b)^2=(a-b)^2$ to $c-b=a-b$ because it assumes the positive root. ✓

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In order to prove something, you may need to use any of your maths skills and knowledge.
It is important to define any algebraic expressions at the start of a proof.
This is so it is clear what cases you are proving.

Pupils may struggle with showing why the false proofs are false.

Checking what each variable represents will help pupils see why each of the 'proofs' do not work.

Function - A function is a mathematical relationship that uniquely maps values of one set to the values of another set.
Expression - An expression contains one or more terms, where each term is separated by an operator.