Definition: Composition of functions means feeding the output of one function into another, written (f o g)(x) = f(g(x)), so g is applied first and f second.
What does composition mean?
The notation (f o g)(x) means f(g(x)). Work from the inside out: evaluate g at x, then use that result as the input of f. Order matters, because f(g(x)) and g(f(x)) are usually different.
How do you build a composite formula?
Replace every x in the outer function with the entire inner expression.
Worked example: let f(x) = 3x – 2 and g(x) = x2 + 5.
- f(g(x)) = 3(x2 + 5) – 2 = 3x2 + 13
- g(f(x)) = (3x – 2)2 + 5 = 9x2 – 12x + 9
At x = 2: f(g(2)) = f(9) = 25, which matches 3(4) + 13 = 25.
What is the typical error?
Multiplying the functions instead of composing them. The dot in f o g is a small circle, not a times sign. Mathbot also warns about domain: the input must be allowed by g, and g’s output must be allowed by f.
Example: f(x) = 3x – 2, g(x) = x^2 + 5: f(g(x)) = 3x^2 + 13
Where to learn it
- Combining and Inverting Functions – lesson
- Domains, Composition and Inverse Functions Explained – lesson