Hi,

In the function notation $f(x) = y, f$ is the name of the function, $x$ is the input to the function and $y$ is the output of the function. An example is the square root function on your calculator. The name of the function is $\sqrt{}.$ If you type a number, say 16 on the keypad then you are saying $x = 16.$ Pushing the $\sqrt{}$ button executes the function and you screen reads 4, which is the output of the function. This $\sqrt{16} = 4.$

If you type $3\times 16 + 1$ press return and then the $\sqrt{}$ the output is 7. That is because $3 \times 16 + 1 = 49$ and hence $\sqrt{3 \times 16 + 1} = \sqrt{49} = 7.$

What about your function $f(x) = 3x + 1?$ The function is described by a formula. $f(x) = 3x + 1$ says multiply the input by 3 and then add one to the result. So what if the input is $s + 5$ where $s$ is some number? The function description says "multiply the input by 3 and then add one to the result" and hence

\[f(s + 5) = 3(s + 5) + 1 = 3s + 15 + 1 = 3s + 16.\]

Now try $f(x + 2).$

You might also want to look at our response to question from Paige and also to a question from Yogita.

Penny