A function is a mathematical relationship that uniquely maps values of one set to the values of another set.

An expression contains one or more terms, where each term is separated by an operator.

Problem solving with functions and proof

Maths

Checking and securing understanding of functions

Defining function notation

Finding the inverse of a function

Writing composite functions

Solving equations involving functions

Solving equations involving composite functions

General algebraic forms for specific number properties

Making conjectures about patterns and relationships

Proving or disproving a statement

Writing a generalised statement about specific number properties

Writing a proof

Logical arguments

Multiple approaches to logical arguments

In this future unit, pupils continue to explore composite functions, thereby building on their work from this unit. Pupils need to understand function notation from this unit as this is used in iteration.

The functions unit requires pupils to write composite functions and pupils will need to draw on their understanding of how to manipulate various algebraic expressions and equations, including algebraic fractions.

In this unit pupils will further develop their understanding of algebraic notation and graphs by learning about functions, their inverses and composite functions.

Functions and proof

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

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

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

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

$$a^2-b^2=ab-b^2$$ 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 $$a^2=a\times{b}$$

$$(c+a)(c-a)$$ 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.

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

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

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

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

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

Starter Quiz

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