Question

Can someone help me with this problem please?

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

a. Find (f*g)(x)

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

Thank you.

Answer 1

(f*g)(x) = f(x) * g(x)
so just multiply the two expressions
Or, maybe you mean
(f◦g)(x) = f(g(x))
If so, then
f(g) = 1/g = 1/(x^2+5x)
As with all polynomials, the domain of g is all reals
The domain of f◦g is all reals except where x^2+5x = 0.
D = (-∞,-5) U (-5,0) U (0,∞)

As for the range, note that
x^2 + 5x = (x + 5/2)^2 - 25/4
That never gets smaller than -25/4
So, the range could be y > -4/25
But y>0 when x>0 or x < -5, so the range is
(-∞,-4/25] U (0,∞)