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Chapter 2: Problem 19

For each piecewise-defined function, find (a) \(f(-5),\) (b) \(f(-1),\) (c)\(f(0),\) and ( \(d\) ) \(f(3)\) See Example 2. $$f(x)=\left\\{\begin{array}{ll} 2+x & \text { if } x < -4 \\ -x & \text { if }-4 \leq x \leq 2 \\ 3 x & \text { if } x > 2 \end{array}\right.$$

### Short Answer

Expert verified

f(-5) = -3, f(-1) = 1, f(0) = 0, f(3) = 9

## Step by step solution

01

## - Evaluate f(-5)

Check which piece of the piecewise function applies to when \(x = -5\). Since \( -5 < -4 \), use the function \(2 + x\). Substitute \(x = -5\) into the equation: \[ f(-5) = 2 + (-5) = 2 - 5 = -3 \]

02

## - Evaluate f(-1)

Check which piece of the piecewise function applies to when \(x = -1\). Since \(-4 \leq -1 \leq 2\), use the function \(-x\). Substitute \(x = -1\) into the equation: \[ f(-1) = -(-1) = 1 \]

03

## - Evaluate f(0)

Check which piece of the piecewise function applies to when \(x = 0\). Since \(-4 \leq 0 \leq 2\), use the function \(-x\). Substitute \(x = 0\) into the equation: \[ f(0) = -0 = 0 \]

04

## - Evaluate f(3)

Check which piece of the piecewise function applies to when \(x = 3\). Since \(3 > 2\), use the function \(3x\). Substitute \(x = 3\) into the equation: \[ f(3) = 3(3) = 9 \]

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Evaluating Functions

Evaluating functions means finding the value of a function for a specific input value. When you evaluate a function, you substitute the given input value (usually represented as x) into the function's formula. This process is done step-by-step to ensure accuracy. For example, if you have a function defined as \(f(x) = x^2\), and you need to find \(f(3)\), you replace x with 3 and compute \(3^2 = 9\). Always pay attention to the function's formula and carefully substitute the input value. Completing these steps accurately is crucial to understanding the behavior of the function.

###### Piecewise-Defined Function

A piecewise-defined function is a function that is not defined by a single equation but by multiple equations, each applying to a specific interval of the input values. These functions are often used to model situations where a rule or relationship changes depending on the input. For example:

\ \(f(x)=\begin{cases} 2+x & \text{if } x < -4 \ -x & \text{if }-4 \leq x \leq 2 \ 3x & \text{if } x > 2 \end{cases}\)

When working with piecewise functions, it is essential to note which interval each piece of the function corresponds to. Evaluating the function at any given point depends on identifying and using the correct equation from the provided pieces.

###### Precalculus

Precalculus is a course that prepares students for calculus by covering various mathematical concepts that are foundational for more advanced studies. It includes the study of functions, including piecewise-defined functions, trigonometry, sequences, and limits. The skills you learn in precalculus, such as function evaluation and understanding different types of functions, are essential as they provide the groundwork for calculus topics such as differentiation and integration. Mastering these areas through practice and problem-solving helps build the confidence needed to tackle more complex mathematical challenges.

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