Hi Paige,

In the notation $y = f(x)$ the letter $f$ identifies the function, $x$ is the input to the function and $y$ is the output of the function. Suppose the input is a number as it often is in mathematics and $f$ is the function that squares whatever input it gets. This function exists on my calculator, it has the symbol $x^2$ on the keyboard. If I type 5 on the keyboard and push the key labeled $x^2$ I get the result 25. Here $x = 5$ and $y = 25$ so $f(5) = 25.$

Since $x = 5, x + 2 = 7$ and hence $f(x + 2) = f(7) = 49.$ But $f(x) = f(5) = 25$ so $f(x) + 2 = 25 + 2 = 27.$ The difference is that $f(x+2)$ says to add 2 to the input and $f(x) +2$ says to add 2 to the output.

I hope this helps,
Penny