To solve the problem $(f \circ g)(19)$, where $f(x) = \ln(4x - 1)$ and $g(x) = (x + 3)^2$, we proceed step by step:

1. Compute $g(19)$: $g(x) = (x + 3)^2$ Substituting $x = 19$: $g(19) = (19 + 3)^2 = 22^2 = 484$

2. Substitute $g(19) = 484$ into $f(x)$: $f(x) = \ln(4x - 1)$ Substituting $x = 484$: $f(484) = \ln(4(484) - 1) = \ln(1936 - 1) = \ln(1935)$

3. Evaluate $\ln(1935)$: Using a calculator or computational tool: $\ln(1935) \approx 7.57$

Final Answer:

$(f \circ g)(19) \approx 7.57$

Let me know if you want further explanation or additional questions! Here are related questions to deepen understanding:

1. How does the domain of $g(x)$ affect the composition $f(g(x))$?
2. What is the significance of the natural logarithm in $f(x)$?
3. Can $f(g(x))$ be expressed symbolically without numerical evaluation?
4. What happens if $g(x)$ results in a value that makes $f(x)$ undefined?
5. How would the process differ if $g(x)$ had a cubic term instead?

Tip: Always check if the composed function $f(g(x))$ is valid within its domain before substituting values!