EQN-03-09

Functions & Function Notation

Evaluate f(x), compose functions, and handle "defined operation" symbols — a favourite GAT-E twist.

~5 min read•Difficulty 2/5

f(x) means "apply the rule f to the input x". To evaluate, substitute the input everywhere x appears.

f (x) = 2 x + 3 \to f (4) = 2 (4) + 3 = 11

The input can itself be an expression.

f (x) = x^{2} - 1 \to f (a + 1) = (a + 1)^{2} - 1

f(g(x)) means do g first, then f.

g (x) = x + 2, f (x) = 3 x \to f (g (5)) = f (7) = 21

The GAT-E often invents a symbol and defines it, e.g. "a ★ $b = a^{2} - b$ ". Just follow the definition.

a ★ b = a^{2} - b \to 5 ★ 3 = 25 - 3 = 22

Read the definition carefully and substitute in the given order.

Worked examples

Example 1 (difficulty 1/5)

f (x) = 3 x - 5,

what is f(6)?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Define a ◇

b = 2 a + b^{2} .

What is 3 ◇ 4 ?

◎ Predict the answer first:

Reading f(3) as

f \times 3

Why: The notation looks like multiplication.

Fix: It means "the value of f when

x = 3

Doing the outer function before the inner in f(g(x))

Why: We read left to right.

Fix: Composition works inside-out: evaluate g first.