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Question
If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2
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Solution
f(x) and g(x) are polynomials.
⇒ f : R → R and g : R → R.
So, fog : R → R and gof : R → R.
(1) (fog) (x) = f (g (x))
= f (x2 + 1)
= 2 (x2+1) +5
=2x2 + 2 + 5
= 2x2 +7
(2) (gof) (x) = g (f (x))
= g (2x +5)
= g (2x + 5)2 + 1
= 4x2 + 20x +26
(3) (fof) (x) = f (f (x))
= f (2x +5)
= 2 (2x + 5)+5
= 4x + 10 + 5
= 4x +15
(4) f2 (x) = f (x) x f (x)
= (2x +5) (2x + 5)
= (2x + 5)2
= 4x2 + 20x +25
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