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प्रश्न
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
उत्तर
f(x)= `sqrt (x + 3)`
For domain, x + 3≥0
⇒ x≥ −3
Domain of f =[-3, ∞)
Since f is a square root function, range of f =[0, ∞)
f : [−3, ∞) → [0, ∞)
g (x)= x2+1 is a polynomial.
⇒ g : R → R
Computation of fog:
Range of g is not a subset of the domain of f.and domain (fog)={ x: x ∈ domain of g and g (x) ∈ domain of f (x) }
⇒ Domain (fog) = { x : x ∈ R and x2+1∈ [−3, ∞)}
⇒ Domain (fog)={ x : x ∈ R and x2+1 ≥−3 }
⇒ Domain (fog)={x : x ∈ R and x2+4 ≥ 0}
⇒ Domain (fog) = {x : x ∈ R and x ∈ R}
⇒ Domain (fog) = R
fog : R → R
(fog) (x) = f(g (x))
= f (x2+1)
= `sqrt(x^2 +1 +3)`
= ` sqrt (x^2 +4)`
Computation of gof :
Range of f is a subset of the domain of g.
gof : [−3, ∞) → R
⇒ (gof) (x) = g (f (x))
=g ` sqrt (x +3)^2 +1`
= x + 3 + 1
= x + 4
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