# Difference between function and function of $x$

For example in $y=3x+4$ what is the function and what is the function of $x$. I don't get the difference between $f$ and $f(x)$.

• This question could lead to a lot of "what is a function" discussion that would not help with the difficulty you actually have. It might help if you edit the question to include a concrete example of where you saw $f$ and $f(x)$ both written in the same sentence (or same paragraph, or otherwise somehow connected) where you don't understand why it has to be $f$ in one place and $f(x)$ in the other. I recommend quoting the confusing text directly; don't try to paraphrase it. – David K Dec 22 '16 at 21:26
• I've often seen confusion between $f$ and $f(x)$ even from experienced mathematicians. It is something you just have to get used to, people getting wrong... – Rasmus Erlemann Dec 22 '16 at 21:40

$f$ is the function defined by $f(x)=3x+4$

while

$f(x)$ is the value of the function $f$ at the point $x$

• the fact that in many cases $f(x)$ is used to indicate $f$ doesn't help :D – Ant Dec 22 '16 at 21:23
• @Ant True! I like what Schwartz started using in distribution theory (perhaps someone before?), namely $f(\hat x)$ for $f$ (when it is really necessary!). For example $e^{\hat x}$ when it is really convenient writing $e^x$ but when it is the function, not a specific value. – John B Dec 22 '16 at 21:26
• Ah interesting, I wasn't aware of that :-) – Ant Dec 22 '16 at 21:30
• If f(x) is a value then why derivative of a function is written like f'(x) – user401345 Dec 23 '16 at 9:43
• The same: it is written $f'$, not $f'(x)$. – John B Dec 23 '16 at 10:03

A function is a relation between two sets (in your case these two sets are the real $x-axis$ and the real $y-axis$) with the property that each $x$ is associated to only one $y$. Functions are usually labelled by letters $f$, $g$, $h$ and so on and you can also write explicitly what is the independent variable of the relation. In your example $f$ is a function that takes any number $x\in \mathbb R$ as input and it gives $3x+4=y\ or\ f(x)\in \mathbb R$ as output, so you can choose any number $x$ (independent variable), put it into $3x+4$ to get the number $y\in\mathbb R$ (dependent variable):

$$f: x\longrightarrow 3x+4=y=f(x)$$

If you want, you can write:

$$f(x): x\longrightarrow 3x+4=y=f(x)$$

so it is clear what the independent variable is, but $f(x)$, called also $y$, usually is the value of the function calculated at some $x$ whereas $f$ is the function i.e. the relation between the sets.