Question

Let f open parentheses x close parentheses equals ln open parentheses x squared close parentheses space space space a n d space space space g open parentheses x close parentheses equals square root of e to the power of 3 x end exponent end root. Find f o g open parentheses x close parentheses and its domain?

Answer 1

`(f\ o\ g)(x) = 3x`

`-\infty < x < \infty`

Explanation:

See comment for right presentation of question

Given

`f(x) = ln(x^2)`

`g(x)=\sqrt{e^(3x)}`

Solving (a): (f o g)(x)

This is calculated as:

`(f\ o\ g)(x) = f(g(x))`

We have:

`f(x) = ln(x^2)`

`f(g(x)) = \ln((g(x))^2)`

Substitute:
`g(x)=\sqrt{e^(3x)}`

`f(g(x)) = \ln(\sqrt{e^(3x)})^2`

Evaluate the square

`f(g(x)) = \ln(e^(3x))`

Using laws of natural logarithm:

`\ln(e^(ax)) = ax`

So:

`f(g(x)) = \ln(e^(3x))`

`f(g(x)) = 3x`

Hence:

`(f\ o\ g)(x) = 3x`

Solving (b): The domain

We have:

`f(g(x)) = 3x`

The above function has does not have any undefined points and domain constraints.

Hence, the domain is:
`-\infty < x < \infty`