Q. 45.0( 1 Vote )

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

(i) fog

(ii) gof

(iii) fof

(iv) f^{2}

Also, show that fof ≠ f^{2}.

Answer :

f(x)= 2x + 5 and g(x)= x^{2 +} 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 (x^{2} + 1)

= 2(x^{2} + 1) + 5

⇒ fog(x)=2x^{2} + 7

Hence fog(x) = 2x^{2} + 7

(ii) gof(x) = g(f(x)) ^{–} = g (2x + 5)

= (2x + 5)^{2} + 1

gof(x)= 4x^{2} + 20x + 26

Hence gof(x) = 4x^{2} + 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) f^{2}(x) = [f(x)]^{2}= (2x + 5)^{2}

= 4x^{2} + 20x + 25

∴ from (iii) and (iv)

fof ≠ f^{2}

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