Domain of Compositions

Pre-Calculus · Axiom Academy

Where is f(g(x)) defined? An input survives only if it clears two gates in a row. Read inside-out. The input x enters g first; its output g(x) then becomes the input to f . For f(g(x)) to be defined, both of these have to hold at once: Gate 1 — the input must be legal for g Gate 2 — the value g(x) must be legal for f 2. Worked Example: A Square Root Gate Take and g(x) = x - 4 . Here g accepts everything , so Gate 1 lets all of x through. The real filtering happens at Gate 2, where f demands a non-negative input. Watch how every x on the line clears Gate 1, but Gate 2 turns away every value below 4 — because there g(x) = x - 4 is negative and rejects it. The survivors are exactly . 3. A Harder Example: Punching Holes Now and g(x) = x^2 - 1 . Again g takes all reals, so Gate 1 is wide open. But f forbids a 0 input, and g(x) hits 0 at two places — so Gate 2 punches out exactly those points. Don't find and stop. Here is all reals — but the composition is still undefined at x = -1 and x = 1 . Notice x = 0 is perfectly fine ( g(0) = -1 , and ): the rejected points are where g(x) = 0 , not where x = 0 . Gate 2 is what catches them. 4. The Domain Is an Intersection Both gates together say: keep the x that satisfy the first condition AND the second. That "and" is a set intersection — the final domain is the overlap of two sets. All x in , so that g(x) even exists. All x for which the output g(x) lands in . Their intersection is the set where f(g(x)) is defined.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.