f(x) = 2x and g(x) = x + 1. Find fg(x) and gf(x). Are they the same?
Step 1 Find fg(x)
fg(x) = f(g(x)) = f(x + 1)
= 2(x + 1) = 2x + 2
Step 2 Find gf(x)
gf(x) = g(f(x)) = g(2x)
= 2x + 1
Step 3 Compare
fg(x) = 2x + 2
gf(x) = 2x + 1
They are NOT the same!
This study notes covers Worked Example 3: Showing Order Matters within Composite Functions for GCSE Mathematics. Revise Composite Functions in Algebra for GCSE Mathematics with 14 exam-style questions and 12 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 6 of 8 in this topic. Use this study notes to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Composite Functions
The composite function fg(x) means: A) Apply g first, then apply f to the result B) Apply f first, then apply g to the result C) Multiply f(x) by g(x) D) Add f(x) to g(x)
Explain why, in general, fg(x) and gf(x) are NOT equal. You may use an example to support your explanation.