Question

Let f(x)= √x+1/3 and g(x)= √x. Find (f * g)(x). Assume all appropriate restrictions to the domain.

A. (f * g)(x)= √x^2+x / 3

B. (f * g)(x)= x+1 / 3

C. (f * g)(x)= x^2+x / 3

D. (f * g)(x)= x+√x / 3

Answer 1

D

Explanation:

Operating with functions f(x) and g(x).

One of the operations is the product: f(x) *g(x) = (fg)(x)

So let's calculate the product of these two functions. The Domain under the radicals must be x

`The \:values\: of \:the\: Domain\: must\: be \:x\geq 0\\\\f(x)=√(x) +(1)/(3) \:and\:g(x)=√(x) \\(f*g)(x)=(√(x) +(1)/(3) )(√(x))=\\(fg)(x)=\sqrt{x^(2) } +(√(x) )/(3) \\(fg)(x)=x+(√(x) )/(3)`

Answer 2

For this case we have the following functions:

`f (x) = \sqrt {x} + \frac {1} {3}\\g (x) = \sqrt {x}`

We must find
`(f * g) (x)`. By definition we have to:

`(f * g) (x) = f (x) * g (x)`

So:

`(f * g) (x) = (\sqrt {x} + \frac {1} {3}) * \sqrt {x}\\(f * g) (x) = (\sqrt {x}) ^ 2+ \frac {1} {3} (\sqrt {x})\\(f * g) (x) = x + \frac {\sqrt {x}} {3}`

Answer:

Option D