Composition of Functions

What is function composition?

Composition means plugging one function into another. The notation $f(g(x))$ (read "f of g of x") means: first apply $g$ to $x$, then apply $f$ to the result. It's like a pipeline — input goes through $g$ first, then the output of $g$ goes through $f$.

Important: $f(g(x))$ and $g(f(x))$ are generally NOT the same. Order matters!

Step-by-step solution

Given $f(x) = x^2 + 1$ and $g(x) = 3x - 2$.

Finding $f(g(x))$

Step 1 — Start with the outer function $f$. $f(\text{something}) = (\text{something})^2 + 1$.

Step 2 — Replace "something" with $g(x) = 3x - 2$:

$f(g(x)) = f(3x - 2) = (3x - 2)^2 + 1$

Step 3 — Expand:

$(3x - 2)^2 + 1 = 9x^2 - 12x + 4 + 1 = 9x^2 - 12x + 5$

Finding $g(f(x))$

Step 1 — Start with the outer function $g$. $g(\text{something}) = 3(\text{something}) - 2$.

Step 2 — Replace "something" with $f(x) = x^2 + 1$:

$g(f(x)) = g(x^2 + 1) = 3(x^2 + 1) - 2$

Step 3 — Simplify:

$= 3x^2 + 3 - 2 = 3x^2 + 1$

Comparison

$f(g(x)) = 9x^2 - 12x + 5 \neq g(f(x)) = 3x^2 + 1$

They are different! For example, at $x = 1$: $f(g(1)) = 9 - 12 + 5 = 2$, but $g(f(1)) = 3 + 1 = 4$.

Numerical verification

Tracing $x = 2$ through both compositions:

• $f(g(2))$: First $g(2) = 3(2) - 2 = 4$. Then $f(4) = 4^2 + 1 = 17$. ✓ Matches $9(4) - 12(2) + 5 = 36 - 24 + 5 = 17$.
• $g(f(2))$: First $f(2) = 4 + 1 = 5$. Then $g(5) = 3(5) - 2 = 13$. ✓ Matches $3(4) + 1 = 13$.

Common mistakes

• Confusing the order. In $f(g(x))$, apply $g$ FIRST, then $f$. The inner function goes first.
• Not fully expanding. $(3x - 2)^2 = 9x^2 - 12x + 4$, not $9x^2 - 4$ or $9x^2 + 4$.

Try it in the visualization

Both compositions are graphed. Trace a specific $x$-value through the pipeline: see it enter $g$, then the output enters $f$. The two resulting curves are visibly different, proving composition isn't commutative.

They're different! Composition is not commutative.