Composite Function | Lesson Note

Composite Functions – Class 10 Additional Maths | New Millennium Academy

Composite Functions

Understanding function composition with real-life examples, diagrams, and step-by-step explanation

Activity 1

"Double a number and add 3" — apply it for 5.

Calculation 1: $$2 \times 5 = 10$$

Calculation 2: $$10 + 3 = 13$$

Composite function diagram for Activity 1

Observation: The initial value of calculation 1 is 5, while the initial value of calculation 2 is 10, which is the result of calculation 1.

Conclusion: Two calculations combine together to give the final result.

Activity 2

Real life analogy with two buses.

Observation:

Bus A takes Ram from Pokhara to Mugling.
Bus B takes Ram from Mugling to Kathmandu.
Ram travels from Pokhara to Kathmandu by using two buses.

Conclusion: Passenger dropped by Bus A is taken by Bus B; the output of the first step becomes the input of the next step.

Activity 3

Function machine analogy for composite functions.

Observation:

Machine 1 takes 4 and gives 16.
Machine 2 takes 16 and gives 32.
From 4 to 32 is the work of two machines together.

Conclusion: The output of one function becomes the input of another; this is called a composite function.

Formal Definition

Composite Function: Let $f: A \rightarrow B$ be a function from set $A$ to $B$ and $g: B \rightarrow C$ be a function from set $B$ to $C$. The function defined from set $A$ to $C$ is called the composite function of $f$ and $g$.

It is denoted by $g \circ f (x)$.

Mathematically: $$(g \circ f)(x) = g(f(x))$$

Why is it written $(g \circ f)(x)$?

In mathematics, function notation is read from right to left for the order of operations. Because $f$ acts on $x$ first, it is written closer to $x$.

In $g(f(x))$, the function $f$ is physically closer to $x$.

Because $f$ acts on $x$ first, it is written on the inside.

The notation $(g \circ f)$ mirrors the nested notation $g(f(x))$.

If we wrote $f \circ g$, it would mean "apply $g$ first and then $f$", which is not the same process when $f$ takes the input first.

Composite Function Diagrams

Composite Function with Ordered Pairs

Let $f = \{(a, p), (b, q), (c, r)\}$ and $g = \{(p, 1), (q, 2), (r, 3)\}$.

Ordered pairs composite function diagram

The outputs of $f$ are the inputs of $g$, so the composite is: $$g \circ f = \{(a,1), (b,2), (c,3)\}$$

$a, b, c$ are inputs of $f$.

$p, q, r$ are outputs of $f$ and inputs of $g$.

$1, 2, 3$ are outputs of $g$.

Solution: $$g(f(a)) = g(p) = 1$$ $$g(f(b)) = g(q) = 2$$ $$g(f(c)) = g(r) = 3$$

Composite Function with Formulas

Let $f(x) = 2x - 1$ and $g(x) = 3x + 5$.

$$ (g \circ f)(x) = g(f(x)) = g(2x-1) = 3(2x-1) + 5 = 6x - 3 + 5 = 6x + 2 $$

$$ (f \circ g)(x) = f(g(x)) = f(3x+5) = 2(3x+5) - 1 = 6x + 10 - 1 = 6x + 9 $$

Conclusion: $(g \circ f)(x) \neq (f \circ g)(x)$, so composite functions are not commutative.

Key Points to Remember

$f^2 = f \circ f$
$g^2 = g \circ g$
For $g \circ f$, domain is the domain of $f$ and range is the range of $g$.

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