Before you read

Answer first — see what you already know.

A temperature conversion uses $f (x) = 3 x + 1$ and a scaling function $g (x) = x^{2}$ . What is $f (g (2))$ ?

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Answer:

Before you read

Answer first — see what you already know.

If $f (x) = x - 5$ and $g (x) = 2 x$ model transformations, what is $g (f (x))$ ?

Check your answer

Answer:

Theory

Operations on Functions

Given two functions $f$ and $g$ , you can create new functions:

These are straightforward — just apply both functions and combine.

Composing f(g(x))

Theory

Function Composition: f(g(x))

Composition is plugging one function INTO another:

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

Read it as: " $f$ of $g$ of $x$ " — first apply $g$ , then apply $f$ to the result.

Important: $f(g(x)) \neq g(f(x))$ in general! Order matters.

Example 1

If $f(x) = 2x + 3$ and $g(x) = x^2$ , find $f(g(2))$ .

Inside first: $g(2) = 2^2 = 4$ .

Then outside: $f(4) = 2(4) + 3 = 11$ .

f(g(2)) = 11

Example 2

If $f(x) = x + 1$ and $g(x) = 3x$ , find $f(g(x))$ and $g(f(x))$ .

$f(g(x)) = f(3x) = 3x + 1$

$g(f(x)) = g(x + 1) = 3(x + 1) = 3x + 3$

Different! $f(g(x)) \neq g(f(x))$ .

f(g(x)) = 3x + 1

g(f(x)) = 3x + 3

Example 3

If $f(x) = \sqrt{x}$ and $g(x) = x + 9$ , find $f(g(x))$ .

$f(g(x)) = f(x + 9) = \sqrt{x + 9}$

f(g(x)) = \sqrt{x + 9}

3 more sections including examples, practice problems, and step-by-step solutions.

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