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Chapter 2: Problem 18

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

### Short Answer

Expert verified

f(-5) = -7, f(-1) = -3, f(0) = -2, f(3) = 2.

## Step by step solution

01

## - Understanding the piecewise function

This function is defined differently based on the value of x. For values of x less than 3, the function is defined as \(f(x) = x - 2\). For values of x greater than or equal to 3, the function is defined as \(f(x) = 5 - x\).

02

## - Find f(-5)

Since \(-5 < 3\), use \(f(x) = x - 2\). Thus, \(f(-5) = -5 - 2 = -7\).

03

## - Find f(-1)

Since \(-1 < 3\), use \(f(x) = x - 2\). Thus, \(f(-1) = -1 - 2 = -3\).

04

## - Find f(0)

Since \(0 < 3\), use \(f(x) = x - 2\). Thus, \(f(0) = 0 - 2 = -2\).

05

## - Find f(3)

Since \(3 \geq 3\), use \(f(x) = 5 - x\). Thus, \(f(3) = 5 - 3 = 2\).

## Key Concepts

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

###### Evaluating Piecewise Functions

Piecewise functions can look a bit tricky at first, because they're defined by different equations depending on the input value. To evaluate these functions, you need to determine which part of the piecewise function applies based on the given input.

For example, in the function provided, f(x)=

If the input is less than 3, you use the equation: x−2. For input values equal to or greater than 3, you use the equation: 5−x. To find f(-5), we see that −5<3, so we use the equation x−2 which gives us . Next, to find f(−1), we also use the same equation since. To find f(0), the same equation is applicable. Finally to find f(3), the condition . This logical approach will help significantly when faced with similar questions.

###### Precalculus Problem Solving

In precalculus, problem solving typically involves understanding and applying different kinds of functions and equations. Piecewise functions, like the one in this exercise, are a crucial part of this.

Gaining proficiency requires practice and a clear methodology. When solving piecewise function problems, always:

- Identify which part of the function applies to the given input value.
- Apply the correct equation to the input value.
- Calculate the result accurately.

Let's apply this in a step-by-step way to the given problem:

- Determine which rule of the piecewise function applies for each input value: -5, -1, 0, and 3.
- For each input, substitute it into the correct equation and perform the calculation.
- Ensure your results make sense within the context of the function definition.

This approach not only applies to piecewise functions but also strengthens your understanding of functions in general, preparing you for more complex precalculus problems.

###### Function Definition Based on Domain

The domain of a function is the set of all possible input values (usually x-values) that the function can take. In a piecewise function, the domain is divided into sections, each with its own formula.

For the function in this exercise:

- f(x)= When −x

This approach ensures that the function is well-defined over its entire domain, even though different rules apply in different regions.

Understanding this is crucial, as it helps you correctly identify which part of the function to use when given any specific x-value.

Always remember that a clear and accurate definition of the domain not only helps in evaluating the piecewise function but also ensures continuity and correctness of the function’s behavior across different segments.

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