Question

Let (f) = 1/x and g(x)=x^2 +5x.

a. Find (f*g)(x)
b. Find the domain and range of (f*)(x)

Answer 1

I will interpret (f*g)(x) as f(g(x))
= f(x^2 + 5x)
= 1/(2x^2 + 5x)

clearly we cannot divide by zero.
When is 2x^2 + 5x equal to zero
x(2x + 5) = 0
x = 0, x = -5/2
so the domain would be any real number except x = 0 and x = -5/2
the range would be any real number, see the graph

https://www.wolframalpha.com/input/?i=plot+y+%3D+1%2F%282x%5E2+%2B+5x%29

Answer 2

Thank you so much :) This actually makes sense too.