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.
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