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Given f(x) = x/(x + 1), g(x) = x¹⁰, and h(x) = x + 3, find the formula for the composite function (f o g o h)(x), state its domain, and compute (f o g o h)(-1).

1. (f o g o h)(x) = (x+3)¹⁰ / ((x+3)¹⁰ + 1), domain x != -4, value = 1024/1025
2. (f o g o h)(x) = (x+3)¹⁰ / ((x+3)¹⁰ - 1), domain x != -2, -4, value = 1024/1023
3. (f o g o h)(x) = (x+3)¹⁰ + 1, domain all real x, value = 1025
4. (f o g o h)(x) = (x+3)¹⁰ / ((x+3)¹⁰ + 1), domain all real x, value = 1024/1025

Correct answer: (f o g o h)(x) = (x+3)¹⁰ / ((x+3)¹⁰ + 1), domain all real x, value = 1024/1025

Solution

h(x) = x + 3. g(h(x)) = (x + 3)¹⁰. f(g(h(x))) = (x+3)¹⁰ / ((x+3)¹⁰ + 1). The only restriction from f(u) = u/(u+1) is u != -1, but u = (x+3)¹⁰ >= 0 always, so the denominator (x+3)¹⁰ + 1 >= 1 > 0 for all real x; domain = all real numbers. At x = -1: (x + 3) = 2, 2¹⁰ = 1024, value = 1024/(1024 + 1) = 1024/1025.

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