Interpreting Function Notation, Graphs, and Context

(Video) Function Notation

by PatrickJMT

Here's a video by PatrickJMT where he explains function notation.

Before, we would express equations in the form: $$y=x^2+x\\ y=8x-5$$ Function notation changes it up a bit, and uses $f(x)$ instead of $y$. It is important to note that this isn't "f times x." Instead, it's read "f of x". $$f(x)=x^2+x\\ g(x)=8x-5$$ Notice that we can even use $g(x)$ instead. This way we have a distinction between the two equations.

If we're told to find $f(4)$, this means we plug in a $4$ into every spot in the equation where there is an $x.$ Since our function is $f(x)=x^2+x,$ then: $$\begin{align} f(4)&=\color{red}{4}^2+\color{red}{4}\\ &=16+4\\ f(4)&=20 \end{align}$$

If we're told to find $g(4),$ we do the same thing with the other function. $$\begin{align} g(4)&=8(\color{red}{4})-5\\ &=32-5\\ g(4)&=27 \end{align}$$

We can even have expressions inside the parentheses. Suppose we want to find $f(x-3)$. This means we put an $x-3$ into all the spots that have an $x.$ $$\begin{align} f(x-3)&=(\color{red}{x-3})^2+(\color{red}{x-3})\\ &=x^2-6x+9+x-3\\ f(x-3)&=x^2-5x+6 \end{align}$$