If a(x) = 3x + 1 and b(x) = squareroot x - 4, what is the domain of (b*a)(x)?

Solution:

In the algebra of the function that is (f ± g)(x), (fg)(x) and (f/g)(x) the domain of the function will be an intersection of the domain of the functions f and g .

Here the domain of the product of the function is asked thus the required domain will be an intersection of the domain of the function a and b.

Given a(x) = 3x + 1 ⇒ Domain, D₁ = R

b(x) = √(x - 4) ⇒ Domain, D₂ = [4, ∞]

Now, Domain of (b*a)(x) = Domain b(x) ⋂ Domain a(x)

= D₂ ⋂ D₁

= [4, ∞] ⋂ R

= [4, ∞]

Example:

If f(x) = (1 + x) and g(x) = √x - 1 then find the domain of (f/g)(x)

Solution:

Given f(x) = 1 + x

⇒ Domain, D₁ = R and

Domain of g(x) = √x - 1 is D₂ = [1, ∞)

Now the domain of (f/g)(x) is D₁ ⋂ D₂ = [1, ∞) but x = 1 makes the denominator g(x) equal to zero. Hence the domain of the function is (1, ∞)

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If a(x) = 3x + 1 and b(x) = squareroot x - 4, what is the domain of (b*a)(x)?

Summary:

If a(x) = 3x + 1 and b(x) = squareroot x - 4, the domain of (b*a)(x) is [4, ∞].