Q. 44.0( 3 Votes )

Let f, g be two real functions defined by and . Then, describe each of the following functions.

i. f + g

ii. g – f

iii. fg

iv.

v.

vi.

vii. f2 + 7f

viii.

Answer :

Given and


We know the square of a real number is never negative.


Clearly, f(x) takes real values only when x + 1 ≥ 0


x ≥ –1


x [–1, ∞)


Thus, domain of f = [–1, ∞)


Similarly, g(x) takes real values only when 9 – x2 ≥ 0


9 ≥ x2


x2 ≤ 9


x2 – 9 ≤ 0


x2 – 32 ≤ 0


(x + 3)(x – 3) ≤ 0


x ≥ –3 and x ≤ 3


x [–3, 3]


Thus, domain of g = [–3, 3]


i. f + g


We know (f + g)(x) = f(x) + g(x)



Domain of f + g = Domain of f Domain of g


Domain of f + g = [–1, ∞) [–3, 3]


Domain of f + g = [–1, 3]


Thus, f + g : [–1, 3] R is given by


ii. f – g


We know (f – g)(x) = f(x) – g(x)



Domain of f – g = Domain of f Domain of g


Domain of f – g = [–1, ∞) [–3, 3]


Domain of f – g = [–1, 3]


Thus, f – g : [–1, 3] R is given by


iii. fg


We know (fg)(x) = f(x)g(x)







As earlier, domain of fg = [–1, 3]


Thus, f – g : [–1, 3] R is given by


iv.


We know




As earlier, domain of = [–1, 3]


However, is defined for all real values of x [–1, 3], except for the case when 9 – x2 = 0 or x = ±3


When x = ±3, will be undefined as the division result will be indeterminate.


Domain of = [–1, 3] – {–3, 3}


Domain of = [–1, 3)


Thus, : [–1, 3) R is given by


v.


We know




As earlier, domain of = [–1, 3]


However, is defined for all real values of x [–1, 3], except for the case when x + 1 = 0 or x = –1


When x = –1, will be undefined as the division result will be indeterminate.


Domain of = [–1, 3] – {–1}


Domain of = (–1, 3]


Thus, : (–1, 3] R is given by


vi.


We know (f – g)(x) = f(x) – g(x) and (cf)(x) = cf(x)




As earlier, Domain of = [–1, 3]


Thus, : [–1, 3] R is given by


vii. f2 + 7f


We know (f2 + 7f)(x) = f2(x) + (7f)(x)


(f2 + 7f)(x) = f(x)f(x) + 7f(x)




Domain of f2 + 7f is same as domain of f.


Domain of f2 + 7f = [–1, ∞)


Thus, f2 + 7f : [–1, ∞) R is given by


viii.


We know and (cg)(x) = cg(x)



Domain of = Domain of g = [–3, 3]


However, is defined for all real values of x [–3, 3], except for the case when 9 – x2 = 0 or x = ±3


When x = ±3, will be undefined as the division result will be indeterminate.


Domain of = [–3, 3] – {–3, 3}


Domain of = (–3, 3)


Thus, : (–3, 3) R is given by


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