Composite functions combine two functions by applying one after the other: fg(x) means apply g first, then apply f to the result. Inverse functions reverse a function's effect: if f maps 3 to 10, then f⁻¹ maps 10 back to 3. Both are Higher-tier GCSE topics that reward careful substitution and a clear head for order of operations.
What is a composite function?
A composite function chains two functions together. The notation fg(x) — read as "f of g of x" — means:
- First apply g to the input x to get g(x).
- Then apply f to that result: f(g(x)).
Think of it like a machine with two stages. You feed x into the first machine (g), take the output, and feed it straight into the second machine (f). The order matters enormously: fg(x) and gf(x) are usually different functions entirely.
How do you calculate fg(x) and gf(x)?
Given: f(x) = 3x + 1 and g(x) = x².
Finding fg(x):
- Apply g first: g(x) = x².
- Apply f to the result: f(x²) = 3x² + 1.
- So fg(x) = 3x² + 1.
Finding gf(x):
- Apply f first: f(x) = 3x + 1.
- Apply g to the result: g(3x + 1) = (3x + 1)².
- Expand: (3x + 1)² = 9x² + 6x + 1.
- So gf(x) = 9x² + 6x + 1.
Since 3x² + 1 ≠ 9x² + 6x + 1, this confirms that fg ≠ gf in general.
How do you evaluate a composite function at a specific value?
There are two equally valid methods for evaluating fg(3), for example.
Method 1 — substitute into the expression: If fg(x) = 3x² + 1, then fg(3) = 3(9) + 1 = 28.
Method 2 — apply stage by stage:
- g(3) = 3² = 9.
- f(9) = 3(9) + 1 = 28.
Both give 28. Method 2 is safer when you have not yet found the combined expression, because it avoids the risk of getting that expression wrong first.
What is an inverse function?
The inverse function f⁻¹ reverses what f does. If f(2) = 7, then f⁻¹(7) = 2. The inverse undoes the original: applying f then f⁻¹ returns you to where you started.
Graphically, the graph of f⁻¹ is the reflection of the graph of f in the line y = x.
Note: f⁻¹(x) is not the same as 1/f(x). The superscript −1 in function notation means "inverse", not "reciprocal".
How do you find f⁻¹(x) algebraically?
Worked example: Find f⁻¹(x) if f(x) = 3x − 4.
- Write the function as an equation: y = 3x − 4.
- Swap x and y: x = 3y − 4.
- Rearrange to make y the subject:
- Add 4: x + 4 = 3y.
- Divide by 3: y = (x + 4)/3.
- Replace y with f⁻¹(x): f⁻¹(x) = (x + 4)/3.
Second example: Find f⁻¹(x) if f(x) = (2x + 1)/(x − 3).
- Write y = (2x + 1)/(x − 3).
- Swap x and y: x = (2y + 1)/(y − 3).
- Multiply both sides by (y − 3): x(y − 3) = 2y + 1.
- Expand: xy − 3x = 2y + 1.
- Collect y terms: xy − 2y = 3x + 1.
- Factorise: y(x − 2) = 3x + 1.
- Divide: y = (3x + 1)/(x − 2).
- f⁻¹(x) = (3x + 1)/(x − 2).
How do you verify that your inverse is correct?
Apply f and then f⁻¹ (or the other way round) — the result must be x.
Check for the first example: f(x) = 3x − 4, f⁻¹(x) = (x + 4)/3.
f⁻¹(f(x)) = f⁻¹(3x − 4) = ((3x − 4) + 4)/3 = 3x/3 = x. ✓
f(f⁻¹(x)) = f((x + 4)/3) = 3 × (x + 4)/3 − 4 = (x + 4) − 4 = x. ✓
| Step | Composite verification |
|---|---|
| f then f⁻¹ | f⁻¹(f(x)) = x ✓ |
| f⁻¹ then f | f(f⁻¹(x)) = x ✓ |
Both checks passing confirms the inverse is correct.
Frequently asked questions
Does every function have an inverse?
Not always — a function must be one-to-one (each output comes from exactly one input) to have an inverse. For example, f(x) = x² is not one-to-one over all real numbers because f(3) = f(−3) = 9. At GCSE you are usually told the domain is restricted to make the function one-to-one before you are asked to find the inverse.
What is the domain and range of an inverse function?
The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. They are swapped. This is a direct consequence of the graph being reflected in y = x — the x-axis and y-axis roles exchange.
Can you find fg(x) when the functions involve fractions?
Yes, but take extra care with the substitution. If f(x) = 1/(x + 1) and g(x) = 2x, then fg(x) = f(2x) = 1/(2x + 1). The input 2x replaces x in f. Write out every step and resist any urge to simplify before the substitution is complete.
How does this topic relate to function notation taught at GCSE?
Function notation f(x) was introduced to give algebra a compact, precise language. Composite and inverse functions are the natural extension: once you can evaluate f(3), you can evaluate f(g(3)) by applying the same idea twice, and you can find f⁻¹(x) by treating the function rule as an equation to rearrange. The underlying skills — substitution and rearranging — are the same ones you use throughout GCSE algebra.
For guided function work at GCSE with Professor Pi, visit aitutors.me.