fx 3x2 4 gxvx hx 1x Determine the following composite func

f(x) = 3x2 + 4 g(x)vx h(x) 1/x Determine the following composite functions. Determine the domain and range of each. (g f)(x) (h f(x) (h g)(x) (f go h)(x)

Solution

f(x) = 3x^2 + 4

g(x) = sqrt x

h(x) = 1/x

a) (fog)(x) = 3 sqrt x + 4

domain is [ 0, infinity )

range is [ 4 , infinity )

b) ( g o f )(x) = sqrt ( 3x^2 + 4 )

domain ( -infinity , + infinity )

range = [ 2 , infinity )

c) (foh)(x) = 3 (1/x)^2 + 4

= 3 / x^2 + 4

domain is (-infinity , 0 ) U ( 0 , infinity )

range is [ 4 , infinity )

d) ( ho f) (x) = 1 / (3x^2 + 4)

domain is (-infinity , + infinity )

range = ( 0, 1/4 )

e) (goh)(x) = sqrt (1/x)

domian is (0, infinity )

range is (0, infinity )

f) (hog)(x) = 1 / sqrt x

domain is (0, infinity )

range is (0, infinity )

g) ( f o g o h)(x) = 3 ( sqrt 1/x)^2 + 4

= 3 *1/x +4

= 3/x + 4

domain is (-infinity , 0 ) U ( 0 , infinity )

range is (-infinity , 4 ) U (4 , infinity )