If $f(2x-3) = 4x-2$, then what is $f(x)$? 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.
 A: Here is where you went wrong:
If $f(x)= $ [some expression involving $x$], then you can indeed say that $f(x+3)$ = [that expression with $x+3$ filled in for $x$
And, if you have $f(x-3)=$ [some expression involving $x$], then you can indeed say that $f(x)=$ [that expression with $x+3$ filled in for $x$]
So far so good.
But, all you have is that $f(\color{red}{2}x-3)=...$. If you want to 'shift' this, you will have to do this relative to $2x$ ... your mistake was to still treat this as if it was just $x$
What to do? Well, treat the $2x$ as a variable, i.e. Say $y=2x$, and rewrite the formula in terms of this $y$. So, in your case, we can say:
$4x-2=2y-2$
OK, and now we can do our normal shift:
$f(y-3)=2y-2$
And thus:
$f(y)=2(y+3)-2=2y+4$
Substituting $y=2x$ back:
$f(2x)=4x+4$
And hence
$f(x)=2x+4$
A: 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. 
A: If $y=2x-3$, what is $x$? We have $x = {1 \over 2} (y+3)$.
If $x = {1 \over 2} (y+3)$, what is $4x-2$? We have $4x-2=2y+4$.
This will give $f(y)$ in terms of $y$.
Explicitly, $f(y) = 2y+4$.
A: If $x\in\mathbb R$, then\begin{align}f(x)&=f\left(2\frac x2\right)\\&=f\left(2\left(\frac x2+\frac32\right)-3\right)\\&=4\left(\frac x2+\frac32\right)-2\\&=2x+4.\end{align}
A: Let $y$ equal to an expression that will make $2y-3=x$ so that we can have $f(x)$.
This can be done if $y=\frac{x}{2}+1.5$
Plug $y$ in:
$f(2y-3)=4y-2$ and substitute the value of $y$ in terms of $x$:
$f(2(\frac{x}{2}+1.5)-3)=f(x)=4(\frac{x}{2}+1.5)-2=2x+4$
A: Correct answer:
Let $y=2x-3$.  Then $f(y)=4x-2=2(2x-3)+4=2y+4.$
Where you went wrong:
If $f(2x-3)=4x-2=2(2x)-2$, then $f(2x)=2(2x+3)-2=4x+4.$
A: 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. 
