Function Composition

Calculus Readiness · Axiom Academy

Chain two functions into one: the output of the inner function becomes the input of the outer. Watch the value pass through both machines. Think of each function as a machine . A number drops into the inner machine g , its output drops straight into the outer machine f , and what falls out the end is f(g(x)) . The key rule: the inner function runs first . Run g first — the inner machine Feed that result into f — the outer machine Now swap the machines . Send the same x = 3 through f first , then g . The pipeline runs in the opposite order — and a different number falls out the end. Composition is not commutative . . Add 4, then double-and-add-1. . Double-and-add-1, then add 4. . The order you stack the machines changes the result. The "+4" lands at a different stage in each pipeline. Run through g first and that 4 gets doubled by f afterward (becoming +8, plus the +1); run through f first and the 4 is just tacked on at the very end. Different timing, different total. 3. The Domain of a Composition A value of x is only allowed if it can survive both machines. Take g(x) = x - 5 feeding , so . The square root rejects anything negative — so x must make the inner output land at zero or above . The rule: x must be in the domain of g , and the output g(x) must be in the domain of f . Here g accepts everything, but demands . The result: means . Inputs below 5 produce a negative number that cannot accept — they get rejected.

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