Question 12: Find the composite functions f[g(x)] and g[f(x)] of the following functions.
a. f(x) = x2 +1, g(x) = 2x
b. f(x) = sinx, g(x) = 1 – x2
c. 
, g(t) = t2
d. 
, 
e. f(x) = sinx, g(x) = 2x + 3
f. f(x) = 1/x, g(x) = tanx
, g(t) = t2
d. , e. f(x) = sinx, g(x) = 2x + 3
f. f(x) = 1/x, g(x) = tanx
Solution
a. f(x) = x2 +1, g(x) = 2x
To find f[g(x)], put the value of g(x) for all x in f(x).
f[g(x)] = (2x)2 + 1 = 4x2 + 1
To find g[f(x)], put the value of f(x) for all x in g(x).
g[f(x)] = 2(x2 + 1) = 2x2 + 2
b. f(x) = sinx, g(x) = 1 – x2
To find f[g(x)], put the value of g(x) for all x in f(x).
f[g(x)] = sin(1 – x2)
To find g[f(x)], put the value of f(x) for all x in g(x).
g[f(x)] = 1 – sin2x = cos2x
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