find f[g(x)] and g[f(x)].

f(x) = ln(x^2+4); g(x)=2x-1

Let's find both \( f(g(x)) \) and \( g(f(x)) \) given the functions:

- \( f(x) = \ln(x^2 + 4) \)
- \( g(x) = 2x - 1 \)

### Step 1: Find \( f(g(x)) \)
To find \( f(g(x)) \), substitute \( g(x) = 2x - 1 \) into \( f(x) \):

\[
f(g(x)) = f(2x - 1) = \ln((2x - 1)^2 + 4)
\]

Now, simplify the expression inside the logarithm:

\[
f(g(x)) = \ln((2x - 1)^2 + 4)
\]

This is the expression for \( f(g(x)) \).

### Step 2: Find \( g(f(x)) \)
To find \( g(f(x)) \), substitute \( f(x) = \ln(x^2 + 4) \) into \( g(x) \):

\[
g(f(x)) = g(\ln(x^2 + 4)) = 2\ln(x^2 + 4) - 1
\]

So,

\[
g(f(x)) = 2\ln(x^2 + 4) - 1
\]

This is the expression for \( g(f(x)) \).

### Summary:

- \( f(g(x)) = \ln((2x - 1)^2 + 4) \)
- \( g(f(x)) = 2\ln(x^2 + 4) - 1 \)

Would you like a more detailed explanation of the steps or any further questions?

### Related Questions:
1. What happens to \( f(g(x)) \) if \( g(x) = x^2 \) instead?
2. How does the derivative of \( f(g(x)) \) differ from that of \( g(f(x)) \)?
3. How would you find the inverse of \( f(x) \)?
4. What are the domain restrictions for both composite functions?
5. How can you expand \( f(g(x)) \) further if the logarithm is required to be in simpler terms?

**Tip:** Always verify the domain of composite functions to ensure both inner and outer functions are defined.

find f[g(x)] and g[f(x)] where f(x) = ln(x^2+4) and g(x) = 2x-1

Learn how to compute composite functions f(g(x)) and g(f(x)) with the given functions f(x) = ln(x^2 + 4) and g(x) = 2x - 1. This problem covers logarithmic functions and algebraic manipulation, suitable for students in Grades 10-12.

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