Question

given: \\( f(x) = \

$$\begin{cases} x - 1, & \\text{if } -4 \\le x < 2 \\\\ -2x, & \\text{if } 2 \\le x \\le 4 \\end{cases}$$

\\)

evaluate the function.

match the values.

\

$$\begin{array}{|c|c|c|} \\hline f(2) & f(6) & f(0) \\\\ \\hline f(-2) & -8 & \\text{undefined} \\\\ \\hline -3 & f(4) & f(-4) \\\\ \\hline -5 & -1 & -4 \\\\ \\hline \\end{array}$$

Explanation:

Identify the piecewise rules

The piecewise function is defined as:

$$ f(x) =
LATEXBLOCK0
$$

Using the Domain of Functions knowledge point, we determine which rule to apply based on the input value \(x\).

Evaluate each given input

We evaluate the function at each of the requested values:

• For \(f(2)\): Since \(x = 2\) lies in the interval \(2 \le x \le 4\), we use the second rule:

$$ f(2) = -2(2) = -4 $$

• For \(f(6)\): Since \(x = 6\) is outside the domain \([-4, 4]\), the function is:

$$ f(6) = \text{Undefined} $$

• For \(f(0)\): Since \(x = 0\) lies in the interval \(-4 \le x < 2\), we use the first rule:

$$ f(0) = 0 - 1 = -1 $$

• For \(f(-2)\): Since \(x = -2\) lies in the interval \(-4 \le x < 2\), we use the first rule:

$$ f(-2) = -2 - 1 = -3 $$

• For \(f(4)\): Since \(x = 4\) lies in the interval \(2 \le x \le 4\), we use the second rule:

$$ f(4) = -2(4) = -8 $$

• For \(f(-4)\): Since \(x = -4\) lies in the interval \(-4 \le x < 2\), we use the first rule:

$$ f(-4) = -4 - 1 = -5 $$

Match values to their results

We pair each function expression with its corresponding evaluated value:

• \(f(2)

ightarrow -4\)

• \(f(6)

ightarrow \text{Undefined}\)

• \(f(0)

ightarrow -1\)

• \(f(-2)

ightarrow -3\)

• \(f(4)

ightarrow -8\)

• \(f(-4)

ightarrow -5\)

Answer:

The matched values for the function are:

• \(f(2) = -4\)
• \(f(6) = \text{Undefined}\)
• \(f(0) = -1\)
• \(f(-2) = -3\)
• \(f(4) = -8\)
• \(f(-4) = -5\)