Question

example 1 graph a piecewise-defined function
to graph a piecewise-defined function, graph each \piece\ separately.
graph $f(x)=\

$$\begin{cases}2x + 4\\text{ if }x\\leq1\\\\-x + 3\\text{ if }x>1\\end{cases}$$

$. state the domain and range.
first, graph $f(x)=2x + 4$ if $x\leq1$.

• create a table for $f(x)=2x + 4$ using values of $x\leq1$.
• because $x$ is less than or equal to 1, place a $\square$ at $\square$ to indicate that the endpoint is included in the graph.
• then, plot the points and draw the graph beginning at (1, 6).

Explanation:

Step1: Determine the endpoint type

For the function \( f(x) = 2x + 4 \) with \( x \leq 1 \), when \( x = 1 \), we calculate \( f(1)=2(1)+4 = 6 \). Since \( x \) is less than or equal to 1, the endpoint at \( x = 1 \) is included. In graphing, an included endpoint is represented by a closed dot (or closed circle).

Step2: Identify the endpoint coordinates

We already found that when \( x = 1 \), \( f(x)=6 \), so the endpoint is at the point \( (1, 6) \).

Answer:

First box: closed dot (or closed circle)
Second box: \( (1, 6) \)