Q. 45.0( 1 Vote )

If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:

(i) fog

(ii) gof

(iii) fof

(iv) f2

Also, show that fof ≠ f2.

Answer :

f(x)= 2x + 5 and g(x)= x2 + 1


The range of f = R and range of g = [1,∞]


The range of f Domain of g (R) and range of g domain of f (R)


both fog and gof exist.


(i) fog(x) = f(g(x)) = f (x2 + 1)


= 2(x2 + 1) + 5


fog(x)=2x2 + 7


Hence fog(x) = 2x2 + 7


(ii) gof(x) = g(f(x)) = g (2x + 5)


= (2x + 5)2 + 1


gof(x)= 4x2 + 20x + 26


Hence gof(x) = 4x2 + 20x + 26


(iii) fof(x) = f(f(x)) = f(2x + 5)


= 2 (2x + 5) + 5


fof(x) = 4x + 15


Hence fof(x) = 4x + 15


(iv) f2(x) = [f(x)]2= (2x + 5)2


= 4x2 + 20x + 25


from (iii) and (iv)


fof ≠ f2


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