EVALUATING THE VALUE OF COMPOSITION FUNCTION FROM THE GIVEN FUNCTIONS

Evaluate the indicated expression assuming that f(x) = √x, g(x) = (x + 1)/(x + 2) , h(x) = |x - 1|.

(i)  (f o g) (4)

(ii)  (f o g) (5)

(iii)  (g o f) (4)

(iv)  (g o f) (5)

(v)  (f o g o h) (0)

(vi)  (h o g o f) (0)

Answer :

(i)  (f o g) (4) :

(f o g) (4) = f[g(4)] ----(1)

g(x) = (x + 1)/(x + 2)

g(4) = (4 + 1)/(4 + 2) = 5/6

Substitute g(4) = 5/6 in (1).

(f o g) (4) = f(5/6)

= √(5/6)

(ii)  (f o g) (5) :

(f o g) (5) = f[g(5)] ----(2)

g(x) = (x + 1)/(x + 2) 

g(5) = (5 + 1)/(5 + 2) = 6/7

Substitute g(5) = 6/7 in (2).

(f o g) (5) = f(6/7)

= √x

= √(6/7)

(iii)  (g o f)(4) : 

(g o f) (4) = g[f(4)] ----(3)

f(x) = √x

f(4) = √4 = 2

Substitute f(4) = 2 in (3).

(g o f) (4) = g(2)

= (2 + 1)/(2 + 2)

= 3/4

(iv)  (g o f) (5) :

(g o f) (5) = g[f(5)] ----(4)

f(x) = √x

f(5) = √5

Substitute f(5) = √5 in (4)

(g o f) (5) = g[√5]

= (√5 + 1)/(√5 + 2)

(v)  (f o g o h) (0) :

(f o g o h) (0) = (f o g) [h(0)] ----(5)

h(x) = |x - 1|

h(0) = |0 - 1| = 1

Substitute h(0) = 1 in (5). 

(f o g o h) (0) = (f o g) (1)

= f[g(1)] ----(6)

g(x) = (x + 1)/(x + 2)

g(1) = (1 + 1)/(1 + 2)  =  2/3

Substitute g(1) = 2/3 in (6). 

= f[2/3]

= √(2/3)

(vi)  (h o g o f) (0) :

(h o g o f) (0) = (h o g) [f(0)] ----(7)

f(x) = √x

f(0) = √0 = 0

Substitute f(0) = 0 in (7). 

  = (h o g) (0)

= h[g(0)] ----(8)

g(x) = (x + 1)/(x + 2)

g(0) = (0 + 1)/(0 + 2)  =  1/2

Substitute g(0) = 1/2 in (8). 

  = h[1/2]

= |1/2 − 1|

= |(1 - 2)/2|

= |-1/2|

= 1/2

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