Question

graph the function.

\
(x) = \

$$\begin{cases} x^2 - 4 & \\text{for } x \\le 2 \\\\ 2x - 4 & \\text{for } x > 2 \\end{cases}$$

\\

part 1 of 5
the first rule defines a parabola with vertex \\((0, -4)\\).

part 2 of 5
since \\(x \le 2\\), the rightmost point \\((2, 0)\\) is graphed as a solid dot.

part 3 of 5
the second rule defines a line with slope \\(2\\) and \\(y\\)-intercept \\((0, -4)\\).

part 4 of 5
since \\(x > 2\\), the select point \\((\quad, \quad)\\) is graphed as an open dot.

Explanation:

Identify the piecewise function rules

The given piecewise function is:

$$ r(x) =
LATEXBLOCK0
$$

We need to analyze the second rule, which applies when \(x > 2\).

Analyze the second rule

Using the Linear Function Properties knowledge point, the second rule \(r(x) = 2x - 4\) defines a line with slope \(2\) and \(y\)-intercept \((0, -4)\).

Determine the boundary point behavior

Using the Piecewise Functions knowledge point, we evaluate the second rule at the boundary value \(x = 2\):

$$ r(2) = 2(2) - 4 = 0 $$

Since the inequality is strict (\(x > 2\)), the leftmost point of this ray is graphed as an open dot at \((2, 0)\).

Answer:

Since \(x > 2\), the <blank>leftmost</blank> point <blank>\((2, 0)\)</blank> is graphed as an open dot.