Solving equations involving composite functions | Maths

Starter quiz

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

quadratic equations

composite functions ✓

multiple functions
If ${\text{f}}(x)=7x−4$ and ${\text{g}}(x)=3x$ evaluate ${\text{gf}}(11)$ .

'219' ✓
Solve $4(x-3)=22$ $x=$ ______.

'8.5' ✓
If ${\text{f}}(x)=6x−15$ solve ${\text{f}}(x)=57$ $x=$ ______.

'12' ✓
If ${\text{f}}(x)=7x−4$ and ${\text{g}}(x)=3x$ which is an expression for ${\text{gf}}(x)$ ?

${\text{fg}}(x)=21x-4$

${\text{fg}}(x)=3(7x-4)$ ✓

${\text{fg}}(x)=7(3x-4)$

${\text{fg}}(x)=7(3x)-4$
Solve $3x^2+3=150$ .

$-{\sqrt{50}}$

$-7$ ✓

$x=3$

$7$ ✓

${\sqrt{50}}$

Exit quiz

For the function ${\text{fg}}(x)$ the $x$ values are in the domain and the values of ${\text{fg}}(x)$ are in the __________.

range ✓

domain

$y$ values
If ${\text{f}}(x)=3x$ and ${\text{g}}(x)=2x+9$ solve ${\text{fg}}(x)=-3$ $x=$ ______.

'-5' ✓
If ${\text{f}}(x)=3x$ and ${\text{g}}(x)=2x+9$ solve ${\text{gf}}(x)=-3$ $x=$ ______.

'-2' ✓
If ${\text{f}}(x)=3x$ and ${\text{g}}(x)=2x+9$ solve ${\text{fg}}(x)={\text{gf}}(x)$ .

$x=3$

$x=6$

$x=-6$

$x=-3$

This equation has no solution. ✓
If ${\text{f}}(x)=2x-7$ what do you notice when we solve ${\text{f}}(x)={\text{f}}^{-1}(x)$ ?

$x=0$

$y=x$

When we input our solution into function the input and output are equal. ✓

There is no solution.
If ${\text{f}}(x)=7-2x$ and ${\text{g}}(x)=x^2$ solve ${\text{gf}}(x)=9$ .

$x=2$ ✓

$x=-2$

$x=1$

$x=-1$

$x=5$ ✓

Worksheet

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Lesson Details

Key learning points

A composite function may have a given output.
It is possible to solve to find the input.
This can be extended to two composite functions that have a common output.

Common misconception

Confusing the order in which functions are evaluated.

Pupils can substitute their answer into the composite function to check their answer. Putting brackets around the functions can help with confirming the order.

Keywords

Function - A function is a mathematical relationship that uniquely maps values of one set to the values of another set.
Equation - An equation is used to show two expressions that are equal to each other.