For each piecewise-defined function, find (a) ƒ(-5), (b) ƒ(-1), (c) ƒ(0), and (d) ƒ(3).

Verified step by step guidance

1. First, understand the piecewise function definition: \[f(x) = \begin{cases} 2 + x & \text{if } x < -4 \\ -x & \text{if } -4 \leq x \leq 2 \\ 3x & \text{if } x > 2 \end{cases}\] This means the function has three different expressions depending on the value of \(x\).
2. For each value of \(x\) given (\(-5\), \(-1\), \(0\), and \(3\)), determine which part of the piecewise function applies by checking the condition for \(x\) in the definition.
3. Evaluate \(f(-5)\): Since \(-5 < -4\), use the first expression \(f(x) = 2 + x\). Substitute \(x = -5\) into this expression.
4. Evaluate \(f(-1)\) and \(f(0)\): Both \(-1\) and \(0\) satisfy \(-4 \leq x \leq 2\), so use the second expression \(f(x) = -x\). Substitute \(x = -1\) and \(x = 0\) respectively.
5. Evaluate \(f(3)\): Since \$3 > 2\(, use the third expression \)f(x) = 3x\(. Substitute \)x = 3$ into this expression.

Key Concepts

Piecewise-Defined Functions

A piecewise-defined function is defined by different expressions depending on the input value's interval. Understanding how to identify which part of the function applies to a given x-value is essential for evaluating the function correctly.

Evaluating Functions at Specific Points

Evaluating a function at a specific point means substituting the given x-value into the correct expression of the function and simplifying. This process requires careful attention to the domain restrictions of each piece.

Inequalities and Interval Notation

Inequalities define the intervals for each piece of the function. Knowing how to interpret and apply inequalities like x < -4, -4 ≤ x ≤ 2, and x > 2 helps determine which formula to use for each input.