Find (f*g)(x) when f(x)=x^2+5x+6 and g(x)=1/x+3

Answer 1

To find (f*g)(x), we need to multiply the functions f(x) and g(x).

Let's first find g(x):
g(x) = 1/x + 3

Now let's find (f*g)(x) by substituting g(x) into f(x):
(f*g)(x) = f(x) * g(x)
= (x^2 + 5x + 6) * (1/x + 3)

To simplify this expression, we can distribute the terms:
= (x^2 + 5x + 6) * 1/x + (x^2 + 5x + 6) * 3
= (x^2/x + 5x/x + 6/x) + (3x^2 + 15x + 18)

Now we can simplify further:
= x + 5 + 6/x + 3x^2 + 15x + 18
= 3x^2 + 21x + 23 + 6/x

Therefore, (f*g)(x) = 3x^2 + 21x + 23 + 6/x.