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Is it possible to write a function $f$ of the variable $x$ $f(x) = 2x$ in the following way? Imagine $u$ is a variable and $u = x + 3$. Can I write the function $f$ as $f(u - 3) = 2 \times (u - 3)$? I have seen this somewhere but I find it odd. For me, what's inside the brackets is the variable of the function, so I would write it like this: $f(u) = 2 \times (u - 3)$.

So $f(x = 1) = 2 \times 1 = 2$

$f(u = 4) = 2 \times (4 - 3) = 2$

Thanks. C.

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    Yes, you can use substitution as you are showing. – 2017-02-19
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    The equality sign would not be put in $( \cdot )$. Instead, you can say "if $x = 1$ then $f(x) = 2 \times 1$." :) – 2017-02-19

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If $x=u-3$ then you can replace $x$ by $u-3$, and vice-versa, anywhere that $x$ or $u-3$ appears, because they are the same thing . So if $x=u-3$ and $f(x)=2x$ then $$f(u-3)=f(x)=2x=2(u-3).$$ If $f(x)=2x$ for all possible values of $x,$ then $f($whatever$)=2($whatever). And if "whatever" happens to be $u-3$, you cannot change it to $u$ on the LHS of the equation.

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    But if u = 4, then f(u)=2×(u−3) = f(4) = 2 x 1 = 2 gives the same answer as f(4-3) = 2 (4 - 3) = 2. This works with all values of u. So what's wrong with writing f(x) = 2x = f(u) = 2 x (u - 3)? I understand now that f(u - 3) = 2 x (u - 3) is fine because as you say, you take whatever is inside the () as argument and then treat the whole thing as that variable. – 2017-02-20
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    x IS u-3. So u IS x+3. So f(u) IS f(x+3), NOT f(x). – 2017-02-20