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.
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 $x$:

$f(2(\frac{x}{2}+1.5)-3)=f(x)=4(\frac{x}{2}+1.5)-2=2x+4$