1. ## Composite Function

How to do these?
The $e^x2-1$ is suppose to be e power of x then x power of 2-1, 2-1 is located above the x.

Find the function f
1. $g:x\rightarrow \ln x$ and $gf:x\rightarrow 2 \ln (x+1)$

2. $g:x\rightarrow e^x$ and $gf:x\rightarrow e^x2-1$

1. $f:x\rightarrow(x+1)^2$

2. $f:x\rightarrow x^2-1$

2. Originally Posted by cloud5
How to do these?
The $e^x2-1$ is suppose to be e power of x then x power of 2-1, 2-1 is located above the x.

Find the function f
1. $g:x\rightarrow \ln x$ and $gf:x\rightarrow 2 \ln (x+1)$

2. $g:x\rightarrow e^x$ and $gf:x\rightarrow e^x2-1$

1. $f:x\rightarrow(x+1)^2$

2. $f:x\rightarrow x^2-1$
1.
$
g(x)=ln (x)
g(f(x))=2ln(x+1)=ln(x+1)^2$
by the rules for logs

So f(x) must be $(x+1)^2$ as g(?)= ln ? so the $?=f(x)=(x+1)^2$

2. $g(x)=e^x$
$g(f(x))=e^{x^2-1}$ so again g(?)= $e^{x^2-1}$ and you should be able to look at that and see that it's $x^2-1$

3. Originally Posted by cloud5
How to do these?
The $e^x2-1$ is suppose to be e power of x then x power of 2-1, 2-1 is located above the x.

Find the function f
1. $g:x\rightarrow \ln x$ and $gf:x\rightarrow 2 \ln (x+1)$

2. $g:x\rightarrow e^x$ and $gf:x\rightarrow e^x2-1$

1. $f:x\rightarrow(x+1)^2$

2. $f:x\rightarrow x^2-1$
1. $g(\textcolor{red}{x}) = \ln{\textcolor{red}{x}}$

$g[f(x)] = 2\ln(x+1)$

$g[\textcolor{red}{f(x)}] = \ln{\textcolor{red}{(x+1)^2}}$

$f(x) = (x+1)^2$

2. $g(\textcolor{red}{x}) = e^{\textcolor{red}{x}}$

$g[\textcolor{red}{f(x)}] = e^{\textcolor{red}{x^2-1}}$

$f(x) = x^2-1$