If \( f(x)=\ln x, g(x)=e^{3 x} \), and \( h(x)=x^{4} \), find the following. (a) \( (f \circ g)(x) \). What is the domain of \( f \circ g \) ? (b) \( (g \circ f)(x) \). What is the domain of \( g \circ f \) ? (c) \( (f \circ g)(5) \) (d) \( (f \circ h)(x) \). What is the domain of \( f \circ h \) ? (e) \( (f \circ h)(e) \)

Let's solve each part step by step. ### (a) \( (f \circ g)(x) \) The composition \( (f \circ g)(x) \) means we will substitute \( g(x) \) into \( f(x) \): \[ (f \circ g)(x) = f(g(x)) = f(e^{3x}) = \ln(e^{3x}) \] Using the property of logarithms, we can simplify this: \[ \ln(e^{3x}) = 3x \] Thus, \[ (f \circ g)(x) = 3x \] **Domain of \( f \circ g \)**: The function \( g(x) = e^{3x} \) is defined for all real numbers \( x \). The output of \( g(x) \) is always positive, which is required for the logarithm function \( f(x) = \ln x \) to be defined. Therefore, the domain of \( f \circ g \) is: \[ \text{Domain of } (f \circ g) = \mathbb{R} \] ### (b) \( (g \circ f)(x) \) Now, we will find \( (g \circ f)(x) \): \[ (g \circ f)(x) = g(f(x)) = g(\ln x) = e^{3 \ln x} \] Using the property of exponents, we can simplify this: \[ e^{3 \ln x} = x^3 \] Thus, \[ (g \circ f)(x) = x^3 \] **Domain of \( g \circ f \)**: The function \( f(x) = \ln x \) is defined for \( x > 0 \). Therefore, the domain of \( g \circ f \) is: \[ \text{Domain of } (g \circ f) = (0, \infty) \] ### (c) \( (f \circ g)(5) \) Now we will evaluate \( (f \circ g)(5) \): \[ (f \circ g)(5) = 3 \cdot 5 = 15 \] ### (d) \( (f \circ h)(x) \) Next, we find \( (f \circ h)(x) \): \[ (f \circ h)(x) = f(h(x)) = f(x^4) = \ln(x^4) \] Using the property of logarithms, we can simplify this: \[ \ln(x^4) = 4 \ln x \] Thus, \[ (f \circ h)(x) = 4 \ln x \] **Domain of \( f \circ h \)**: The function \( f(x) = \ln x \) is defined for \( x > 0 \). Therefore, the domain of \( f \circ h \) is: \[ \text{Domain of } (f \circ h) = (0, \infty) \] ### (e) \( (f \circ h)(e) \) Finally, we will evaluate \( (f \circ h)(e) \): \[ (f \circ h)(e) = 4 \ln(e) = 4 \cdot 1 = 4 \] ### Summary of Results - (a) \( (f \circ g)(x) = 3x \), Domain: \( \mathbb{R} \) - (b) \( (g \circ f)(x) = x^3 \), Domain: \( (0, \infty) \) - (c) \( (f \circ g)(5) = 15 \) - (d) \( (f \circ h)(x) = 4 \ln x \), Domain: \( (0, \infty) \) - (e) \( (f \circ h)(e) = 4 \)