2.2: Graphs of Functions

In Section 1.9, we defined the graph for an equation.

We now do the same for a function.

The graph of a function with domain $A$ is the set of all possible coordinates $(x, f(x))$ which make the equation true.

The only difference between this definition and the one in Section 1.9 is $y$ was replaced with $f(x)$:

Insight: $f(x)$ is the height from the $x$-axis to the graph.

We have two interpretations of $f(x)$:

  1. Algebraically, $f(x)$ is the output.
  2. Geometrically, $f(x)$ is the height of the function.

Graphing a function is the same exact method introduced in Section 1.9! Except, $y$ is replaced with $f(x)$.

Graph the function $f(x) = \sqrt{x}$.

Graphing Piecewise Functions

Be careful when graphing piecewise functions.

When you see a $<$ or a $>$, plug the number that it should be equal to and draw an open circle.

Graph the function \[f(x) = \begin{cases} x^2 & x \leq 1 \\ 2x + 1 & x > 1\end{cases}\]

Don't forget to plug in $1$ into $2x + 1$, and draw an open circle there. That is where the line $2x + 1$ ends.

Graph the function \[f(x) = \begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0\end{cases}\]

Vertical Line Test

A function has one output for each input.

The graph tells you whether this is true.

Vertical Line Test
A curve in the plane is the graph of a function if no vertical intersects the curve more than once.
Which of the following curves are graphs of functions?