IGCSE Functions and Transformations
Use function notation, compose and invert simple functions, then describe graph transformations.
🎯 Key idea: $f(x)$ is an output, not multiplication. Read it as “the function $f$ of input $x$”.
🔁 1. Function notation
If $f(x)=2x+5$, substitute the input inside the rule.
Worked example: Find $f(-3)$.
$f(-3)=2(-3)+5=-1$.
🔗 2. Composite and inverse functions
For a composite, apply the inside function first: $fg(x)$ means $f(g(x))$.
Worked example: $f(x)=x+4$ and $g(x)=3x$. Find $fg(2)$.
$g(2)=6$, then $f(6)=10$.
So $fg(2)=10$.
Worked example: Find the inverse of $f(x)=3x-2$.
Write $y=3x-2$, swap $x$ and $y$, then rearrange.
$x=3y-2$, so $y=[[frac{x+2}{3}]]$.
$f^{-1}(x)=[[frac{x+2}{3}]]$.
🧭 3. Graph transformations
For $y=f(x)+a$, move the graph up $a$. For $y=f(x-a)$, move it right $a$.
Worked example: Describe $y=(x-2)^2+3$ from $y=x^2$.
Move right $2$ and up $3$.
Matching game: Draw $y=x^2$, then match these descriptions to sketches: $y=x^2+2$, $y=(x-2)^2$, $y=(x+3)^2$. Explain the change before checking with graphing software.
Practice Problems
Sketch or calculate before opening each answer.
1. If $f(x)=4x-1$, find $f(5)$.
Show answer
$f(5)=4(5)-1=19$.
2. $f(x)=x+2$, $g(x)=2x$. Find $fg(3)$.
Show answer
$g(3)=6$, then $f(6)=8$.
Answer: $8$.
3. Describe the transformation from $y=x^2$ to $y=(x+4)^2-1$.
Show answer
Answer: left $4$, down $1$.
Interactive Quiz
Choose an answer, then submit the quiz.
1. If $f(x)=x^2+1$, what is $f(3)$?
2. If $f(x)=x+1$ and $g(x)=2x$, what is $fg(4)$?
3. From $y=x^2$ to $y=x^2-5$ is a move:
Function Explorer
Choose two function families, transform them, and compare their values and compositions at an input.