Mathematics

Problem

Let

f: \mathbb R \to \mathbb R

be defined by

f(x)=2x-3

and

g: \mathbb R \to \mathbb R

be defined by

g(x)=x^{2}+4

. a. (i) Find

(g\circ f)(x)

and simplify. b. (ii) Find

(f\circ g)(x)

and simplify. c. (iii) Evaluate

(g\circ f)(2)

and

(f\circ g)(2)

Intuitive explanation

Function composition is about chaining operations in the right order—when you see $f(g(x))$ , you're plugging the entire output of $g$ into $f$ as its input. The key is to substitute carefully: replace every $x$ in $f$ 's formula with the expression $g(x)$ , then simplify. This builds intuition for how functions nest and transform values step by step.

Intuitive explanation

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