🔢 Mastering Functions: From Notation to Inverses!

Functions sit at the heart of **nearly every algebra problem** you will meet at O-Level and beyond. Understanding their notation, how to chain them together (composite functions) and how to "undo" them (inverse functions) gives you a powerful toolkit for tackling equations, modelling real-world situations and scoring top marks in the Functions section of Paper 2.

Key Concepts to Remember

A function is a rule that assigns **exactly one output** to every valid input. In exam notation, we usually write

f:x\mapsto 2x+3

f(x)=2x+3

. Below are the essentials you must know:

Notation: $f(x)$ means “apply rule $f$ to $x$ .” If $f: x\mapsto 2x-1$ , then $f(4)=7$ .
Composite functions: $fg(x)$ means “apply $g$ first, then $f$ .” Algebraically, $fg(x)=f(g(x))$ .
Inverse functions: $f^{-1}(x)$ reverses $f$ . If $f$ sends 3 to 9, then $f^{-1}$ sends 9 back to 3. A function has an inverse only if it is **one-to-one (bijective)**.
Graphs: $f^{-1}(x)$ is the reflection of $f(x)$ in the line $y=x$ .

Worked Example: Composite & Inverse in Action

Problem: Let

f(x)=3x-2

and

g(x)=\dfrac{x+5}{2}

. (a) Find an expression for

fg(x)

. (b) Determine

f^{-1}(x)

Step 1:
$fg(x)=f\bigl(g(x)\bigr)=f\left(\frac{x+5}{2}\right)=3\left(\frac{x+5}{2}\right)-2$
Step 2:
$=\frac{3(x+5)}{2}-2=\frac{3x+15}{2}-\frac{4}{2}=\frac{3x+11}{2}$
Step 3: To find $f^{-1}(x)$ , start with $y=3x-2$ and swap $x$ and $y$ :
$x=3y-2$
Step 4: Solve for $y$ :
$3y=x+2 \implies y=\frac{x+2}{3}$
, so
$f^{-1}(x)=\frac{x+2}{3}$

Now that you have the **road-map**—notation, composition, and inversion—practise by picking any two simple functions, finding

gf(x)

and both inverses. Repetition cements speed, and speed wins marks. Keep solving and let every function become your friend!