Composite functions

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A composite function applies one function to the output of another. f(g(x)): first apply g to x, then apply f to the result. Notation: (f ∘ g)(x) = f(g(x)) — read "f of g of x."

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(f ∘ g)(x) = f(g(x)). g is applied first (the inner function), then f is applied to the result (the outer function). Think of it like nested brackets — work from the inside out.

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Step 1: g(x) = 3x (the inner function). Step 2: f(g(x)) = f(3x) = (3x) + 2 = 3x + 2. Substitute g(x) = 3x wherever x appears in f.

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Composition (f ∘ g) is not multiplication. f(g(x)) means "substitute g(x) into f" — apply one function to the output of the other. f(x) × g(x) means multiply the two outputs — a completely different operation.

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Step 1: Calculate the inner function first — find g(a). Step 2: Substitute that result into f — find f(g(a)). Always work inside out: inner function first, outer function second.

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Step 1: g(3) = 3² = 9. Step 2: f(g(3)) = f(9) = 2(9) + 1 = 19.

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Step 1: f(5) = 5 − 4 = 1. Step 2: g(f(5)) = g(1) = 3(1) + 2 = 5. Note: this asks for g(f(5)), so f is applied first, then g.

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They applied the functions in the wrong order. For f(g(4)): compute the inner function g(4) first, then substitute into f. The function written on the right (inside the bracket) is always applied first.

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Step 1: Write out g(x). Step 2: Substitute g(x) into f — replace every x in f(x) with the expression g(x). Step 3: Simplify if possible.

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g(x) = x². f(g(x)) = f(x²) = 2(x²) + 3 = 2x² + 3.

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f(x) = x − 1. g(f(x)) = g(x − 1) = 3(x − 1) = 3x − 3.

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They applied the wrong rule. f(x) = (x + 1)² means: take the input, add 1, then square. f(g(x)) = (g(x) + 1)² — substitute g(x) for x throughout. Always replace every x in f with the full expression g(x), including inside brackets.

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No — in general f(g(x)) ≠ g(f(x)). Counterexample: f(x) = x + 1, g(x) = x². f(g(x)) = x² + 1. g(f(x)) = (x + 1)² = x² + 2x + 1. These are different.

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f(g(2)): g(2) = 5, then f(5) = 25. g(f(2)): f(2) = 4, then g(4) = 7. f(g(2)) = 25 ≠ g(f(2)) = 7. The order of composition matters.

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f and g are inverse functions of each other: g = f⁻¹ (and f = g⁻¹). Each function "undoes" the other. Example: f(x) = 2x + 1 and g(x) = (x − 1)/2 satisfy f(g(x)) = x and g(f(x)) = x.

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Read carefully: g(f(x)) means "f is inside g" — apply f first, then g. Memory check: the function closest to x (written on the right) is always applied first. In g(f(x)): f is closer to x → f goes first → then g.