Key takeaway: The composition of functions feeds the output of one function into another, written (f∘g)(x) = f(g(x)), so the inner function g acts first and f acts second.
Grade 11 – Algebra 2F-BF.A.1cFunctions and Graphs
Definition of Composition of Functions
The composition of functions chains two rules into one. You start with an input, run it through the first rule, and hand that result straight to the second rule. The final output depends on the original input only, so the pair behaves like a single new function.
Picture a bakery line with an oven and then a slicer. Swapping the stations usually changes the product. That is why the order matters in any composition of functions, and why \(f\circ g\) differs from \(g\circ f\) in most cases.
Formula and notation
The symbol \(\circ\) is read “of” or “composed with”:
\[(f\circ g)(x)=f\bigl(g(x)\bigr).\]
Work from the inside out: evaluate \(g(x)\), then give that number to \(f\). Allowed inputs belong to the domain of \(g\) and must produce outputs that \(f\) accepts, which restricts the composition of functions.
Key facts
- The function written closest to x is the one that runs first.
- Swapping the order of the two rules usually produces a different new rule.
- The identity function leaves any function unchanged when composed with it.
- Two functions are inverses exactly when both compositions return x.
Composition of Functions example
Let \(f(x)=5x+2\) and \(g(x)=x^2-6\). Find \((f\circ g)(x)\) and \((g\circ f)(x)\), then compare them at \(x=2\).
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Put \(g\) inside \(f\): \((f\circ g)(x)=5(x^2-6)+2=5x^2-28\).
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Put \(f\) inside \(g\): \((g\circ f)(x)=(5x+2)^2-6=25x^2+20x-2\).
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At \(x=2\): the first gives \(20-28=-8\), the second gives \(100+40-2=138\).
The two orders disagree: -8 versus 138 at \(x=2\).
Common mistake
Watch out
Rafael reads \((f\circ g)(2)\) left to right: \(f(2)=12\), then \(g(12)=138\).
Why it is wrong
In \(f(g(2))\) the inner function \(g\) runs first: \(g(2)=-2\), then \(f(-2)=-8\). Rafael actually computed \(g(f(2))\), the other composition of functions.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Functions and Graphs