If (x)=ln x,g(x)=e^3x, and h(x)=x^8,find the following

Question

a)(f*g)(x). What is the domain of f*g?

b)(g*f)(x).What  is the domain  of g*f?

Answer 1

• For the function f + g, f - g, f * g, the domains are defined as the inrersection of the domains of f and g.
• For f/g, the domains is the intersection of the domains of f and g except for the points where g(x) = 0

(a).

The functions are f(x) = ln(x) and g(x) = e^3x and h(x) = x⁸.

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

               = ln(x) * e^3x

The domain of f(x) = ln(x) is (0, +∞)

The domain of g(x) = e^3x is (-∞, +∞)

The domain of (f * g)(x) is (0, +∞) ∩ (-∞, +∞) = (0, +∞).

 

(b).

The functions are f(x) = ln(x) and g(x) = e^3x and h(x) = x⁸.

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

               = e^3x * ln(x)

The domain of f(x) = ln(x) is (0, +∞)

The domain of g(x) = e^3x is (-∞, +∞)

The domain of (f * g)(x) is (0, +∞) ∩ (-∞, +∞) = (0, +∞).