Function notation is a shorthand way to describe how a function maps inputs to outputs.

The expression $f(x)$ means "the value of the function $f$ when the input is $x$." It's a more descriptive way to write $y$ in terms of $x$ and helps us deal with more complex expressions in calculus.

For example, if:

this tells us that:

• The function is named $f$
• For every input $x$, the output is given by the rule $2x + 1$

So $f(3) = 2(3) + 1 = 7$.

This notation lets us plug in values, graph functions, and perform operations on them.

In calculus, function notation is essential because we often study how $f(x)$ changes with $x$, especially as $x$ gets close to some value, which brings us to limits.