Question

question
graph the following function on the axes provided.
$f(x) = \

$$\begin{cases} 4x + 21 & \\text{for } x \\leq -4 \\\\ 4 & \\text{for } x > 1 \\end{cases}$$

$
\boxed{line} \boxed{closed circle} \boxed{open circle}
click and drag to make a line. click the line to delete it.
click on an endpoint of a line to change it.
graph axes with y - axis labeled 6, 7, 8, 9, 10

Explanation:

Step1: Analyze the first piece ($x \leq -4$)

The first piece is $f(x) = 4x + 21$ for $x \leq -4$. This is a linear function. Let's find two points. When $x = -4$, $f(-4) = 4(-4) + 21 = -16 + 21 = 5$. So we have the point $(-4, 5)$ (closed circle since $x \leq -4$ includes $x = -4$). Let's find another point, say $x = -5$: $f(-5) = 4(-5) + 21 = -20 + 21 = 1$. So the point $(-5, 1)$. We can draw a line through $(-5, 1)$ and $(-4, 5)$ (with a closed circle at $(-4, 5)$) extending to the left (since $x \leq -4$).

Step2: Analyze the second piece ($x > 1$)

The second piece is $f(x) = 4$ for $x > 1$. This is a horizontal line. For any $x > 1$, $y = 4$. So we can take a point like $(2, 4)$ (open circle at $x = 1$ since $x > 1$ does not include $x = 1$) and draw a horizontal line to the right from there (with an open circle at $(1, 4)$ and the line going through $(2, 4)$, $(3, 4)$, etc.).

Answer:

To graph the piecewise function:

• For \( f(x) = 4x + 21 \) ( \( x \leq -4 \) ): Plot the point \( (-4, 5) \) (closed circle) and another point (e.g., \( (-5, 1) \)), then draw a line through these points extending left.
• For \( f(x) = 4 \) ( \( x > 1 \) ): Plot an open circle at \( (1, 4) \) and draw a horizontal line to the right through points like \( (2, 4) \), \( (3, 4) \), etc.