Question: I have this statement: 

> 
> If $f(2x-3) = 4x-2,$ the function $f(x)$ is ...?
> 
> 
> 

My attempt was:

Move the function $3$ units to the left

$f(2x) = 4(x+3) -2 = 4x+10$

Divide $x$ by $2$

$f(x) = 2x+10$

Verifiy $f(x) = 2x+10 \to f(2x) = 4x+10 \to f(2x-3) = 4(x-3)+10 = \underbrace{4x-2}\_{f(2x-3)}$

But according to the guide the correct answer is $2x+4$ and i don't know why. Thans in advance.

Answer: 
Put $y=2x-3$. Then $x=(y+3)/2$. In particular
$$
f(y)=4x-2=4\left(\frac{y+3}{2}\right)-2=2(y+3)-2=2(y+2)
$$
as desired.
Since the argument and output are linear, we can try to *extract* $2x-3$ from $4x-2$ through arithmetic means. 

\begin{align}
f(\color{red}{2x-3})&=4x-2\\
&=4x\color{blue}{-6+6}-2\\
&=2(\color{red}{2x-3})+4
\end{align} 

Therefore, $f(x)=2x+4$ as required.