Composition of Functions

MATFunctions

The composition $g \circ f$ feeds the output of $f$ into $g$ , giving $(g \circ f)(x) = g(f(x))$ . It is examined to test order-sensitivity (composition is generally not commutative), to build the single-step function behind chained real-world processes (discount then GST), and as the engine for verifying inverses.

InteractiveThis topic has a hand-built visualisation (composition-flow-diagram.html). It is not wired into the app yet.

The key skills are evaluating compositions, respecting the domain/range matching, and using composition to confirm $f^{-1}$ .

Definition of composition

(g \circ f)(x) = g\big(f(x)\big)

Apply

f

first, then

g

; needs range of

f \subseteq

domain of

g

Not commutative

f \circ g \ne g \circ f \quad (\text{in general})

e.g.

f(x)=2x+3,\ g(x)=x^2+1

give

(f\circ g)(x)=2x^2+5

but

(g\circ f)(x)=(2x+3)^2+1

Associative

(f \circ g) \circ h = f \circ (g \circ h)

Composition is always associative when the domains/ranges match up.

Inverse of a composition

(g \circ f)^{-1} = f^{-1} \circ g^{-1}

The order REVERSES; valid when

f

and

g

are bijections.

Identity under composition

f \circ I_A = f = I_B \circ f

For

f:A\to B

I_A,I_B

are the identity functions on

A,B

with

I(x)=x

; they leave

f

unchanged.

Read $g \circ f$ right-to-left: $f$ acts first, then $g$ acts on that result.
$(f \circ g)(x)$ and $(g \circ f)(x)$ are usually different functions; always compute the requested order and do not assume equality.
To verify $g = f^{-1}$ , show BOTH $(f \circ g)(x)=x$ and $(g \circ f)(x)=x$ ; one direction is not enough.
For chained processes, model each step as a function and compose: a $20\%$ discount $f(p)=0.8p$ followed by $18\%$ GST $g(c)=1.18c$ give $(g \circ f)(p)=0.944\,p$ , so a list price of $2000$ gives $\text{Rs }1888$ .
The inverse of a composition reverses the order: $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$ .
Composition requires the range of the inner function to fit inside the domain of the outer function, or the composition is undefined.
If $f$ and $g$ are both bijections, then $g \circ f$ is also a bijection (hence invertible).

Where the marks go

Applying the functions in the wrong order — students often compute $f(g(x))$ when $g \circ f$ (i.e. $g(f(x))$ ) was asked.
Assuming $f \circ g = g \circ f$ ; verify with the actual rules, since composition is generally non-commutative.
Writing $(g \circ f)^{-1} = g^{-1} \circ f^{-1}$ instead of the correct reversed order $f^{-1} \circ g^{-1}$ .
Checking only $(f \circ g)(x)=x$ when proving an inverse and forgetting the second identity $(g \circ f)(x)=x$ .

How the board asks it

Numericalthe definition of composition
If $f(x) = 2x + 3$ and $g(x) = x^2 - 1$ , find $(g \circ f)(x)$ and $(f \circ g)(x)$ , and hence evaluate $(g \circ f)(2)$ .
Give reasonscomposition is not commutative
Let $f(x) = x + 1$ and $g(x) = 2x$ . Show that $f \circ g \neq g \circ f$ , and state, with reasons, whether composition of functions is commutative in general.
Derive / provecomposition as the engine for verifying inverses
If $f : \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = \dfrac{3x + 4}{5}$ and $g(x) = \dfrac{5x - 4}{3}$ , show that $(f \circ g)(x) = (g \circ f)(x) = x$ , and hence prove that $g = f^{-1}$ .
Applicationchained processes such as discount then GST
A shop offers a $20\%$ discount on the list price, after which $18\%$ GST is charged. Modelling each step as a function, write the single composite function for the final price and hence find the amount payable on an item listed at $\text{₹}2500$ .
Numericalthe inverse of a composition reverses order
If $f(x) = x^3$ and $g(x) = x + 2$ are bijections on $\mathbb{R}$ , find $(g \circ f)^{-1}(x)$ and verify your answer using the rule $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ .
Multiple choicedomain and range matching
If $f(x) = \sqrt{x}$ with $x \geq 0$ and $g(x) = x - 4$ , then $(f \circ g)(x)$ is defined only when: (a) $x \geq 0$ (b) $x \geq 4$ (c) $x \leq 4$ (d) all $x \in \mathbb{R}$ .

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.