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) $$ (f \circ h)(e) $$

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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) $$ (f \circ h)(e) = 4 $$
(a) $$ (f \circ g)(x) = 3x $$, Domain: All real numbers (b) $$ (g \circ f)(x) = x^3 $$, Domain: $$ x > 0 $$ (c) $$ (f \circ g)(5) = 15 $$ (d) $$ (f \circ h)(x) = 4 \ln x $$, Domain: $$ x > 0 $$ (e) $$ (f \circ h)(e) = 4 $$

If , and , find the following.
(a) . What is the domain of ?
(b) . What is the domain of ?
©
(d) . What is the domain of ?
(e)

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