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Let f(x) = sin x and g(x) = 1 - sqrt(x). Determine the domain and range of each composite function: (i) f o g, (ii) g o f, (iii) f o f, (iv) g o g.

1. (i) dom x>=0, range [-1,1]; (ii) dom all real x, range [1 - sqrt of (sin x where sin x>=0)...]; (iii) dom all real, range [-1,1]; (iv) dom 0<=x<=1, range [0,1]
2. (i) dom x>=0, range [-1,1]; (ii) dom {x: sin x >= 0}, range [1 - 1, 1] = [0,1]; (iii) dom all real x, range [-sin1, sin1]; (iv) dom 0<=x<=1, range [1 - 1, 1] = [0,1]
3. (i) dom all real x; (ii) dom x>=0; (iii) dom x>=0; (iv) dom all real x
4. (i) dom x>=0, range [sin(1) form]; (ii) dom all real; (iii) dom [-1,1]; (iv) dom [0,1]

Correct answer: (i) dom x>=0, range [-1,1]; (ii) dom {x: sin x >= 0}, range [1 - 1, 1] = [0,1]; (iii) dom all real x, range [-sin1, sin1]; (iv) dom 0<=x<=1, range [1 - 1, 1] = [0,1]

Solution

(i) f o g = sin(1 - sqrt x); domain x >= 0; the argument 1 - sqrt x ranges over (-inf, 1], covering more than 2 pi so sin attains all of [-1,1]. (ii) g o f = 1 - sqrt(sin x); need sin x >= 0; sin x in [0,1] so sqrt in [0,1], giving 1 - that in [0,1]. (iii) f o f = sin(sin x); sin x in [-1,1] so output in [-sin1, sin1]; domain all reals. (iv) g o g = 1 - sqrt(1 - sqrt x); need x >= 0 and 1 - sqrt x >= 0 => 0 <= x <= 1; output in [0,1].

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