if f(x)=lnx, g(x)=e^3x, and h(x)=x^4 find following
A)(f of g)(x)and the domain of f of g
B) (g of f)(x) and the domain of g of f
C) (f of h)(x) and the domain of f of h
To find the compositions (f of g)(x), (g of f)(x), and (f of h)(x), we need to substitute one function into another function. Let's calculate the compositions step by step:
A) (f of g)(x) and the domain of (f of g)
To find (f of g)(x), substitute g(x) into f(x):
(f of g)(x) = f(g(x)) = ln(g(x)) = ln(e^3x)
Using the property ln(e^x) = x, we can simplify further:
(f of g)(x) = ln(g(x)) = ln(e^3x) = 3x
The domain of (f of g) is the same as the domain of g(x), which is the set of all real numbers. So, the domain of (f of g)(x) is also the set of all real numbers.
B) (g of f)(x) and the domain of (g of f)
To find (g of f)(x), substitute f(x) into g(x):
(g of f)(x) = g(f(x)) = g(ln(x)) = e^3(ln(x))
Using the property e^(ln(x)) = x, we can simplify further:
(g of f)(x) = g(ln(x)) = e^3(ln(x)) = x^3
The domain of (g of f) is the same as the domain of f(x), which is x > 0 (since ln(x) is only defined for x > 0). Therefore, the domain of (g of f)(x) is also x > 0.
C) (f of h)(x) and the domain of (f of h)
To find (f of h)(x), substitute h(x) into f(x):
(f of h)(x) = f(h(x)) = f(x^4) = ln(x^4)
Using the property ln(x^k) = k ln(x), we can simplify further:
(f of h)(x) = f(x^4) = ln(x^4) = 4 ln(x)
The domain of (f of h) is the same as the domain of h(x), which is the set of all real numbers. So, the domain of (f of h)(x) is also the set of all real numbers.