Question

Suppose f(x) = ln(x) + 3 and g(x) = e^3x. Find (f o g)(x) and (g o f )(x) in the
simplest form.

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: f(x) = ln(x) + 3$

and

$\rm \: g(x) = {e}^{3x}$

Now, Consider

$\rm \: fog(x)$

$\rm \: = \: f[g(x)]$

$\rm \: = \: f({e}^{3x})$

$\rm \: = \: ln({e}^{3x}) + 3$

We know,

$\boxed{\tt{ \: \: ln({e}^{x}) = x \: \: }}$

So, using this result, we get

$\rm \: = \: 3x + 3$

$\rm \: = \: 3(x + 1)$

Hence,

$\rm\implies \:\rm \: fog(x) = 3(x + 1)$

Now, Consider

$\rm \: gof(x)$

$\rm \: = \: g[f(x)]$

$\rm \: = \: g(lnx + 3)$

$\rm \: = \: {e}^{3(ln(x) + 3)}$

$\rm \: = \: {e}^{3ln(x) + 9}$

$\rm \: = \: {e}^{3ln(x)} \times {e}^{9}$

We know,

$\boxed{\tt{ \: y \: ln(x) = ln ({x}^{y}) \: }}$

So, using this, we get

$\rm \: = \: {e}^{ln( {x}^{3} )} \times {e}^{9}$

We know

$\boxed{\tt{ \: {e}^{ln(x)} = x \: }}$

So, using this, we get

$\rm \: = \: {x}^{3} \: {e}^{9}$

Hence,

$\rm\implies \:\rm \:gof(x) = \: {x}^{3} \: {e}^{9}$

Thus,

$\rm\implies \:\rm \: fog(x) = 3(x + 1)$

and

$\rm\implies \:\rm \:gof(x) = \: {x}^{3} \: {e}^{9}$

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Basic Concept

f o g is a composite function which means g(x) function is in f(x).

g o f is a composite function which means f(x) function is in g(x).

Answer 2

Refer the Given Attachment