Problem 22 - Algebra and Trigonometry

Question

Evaluate the piece wise defined function at the indicated values. $$ \begin{array}{ll}{f(x)=\left\\{\begin{array}{ll}{5} & {\text { if } x \leq 2} \\\ {2 x-3} & {\text { if } x>2}\end{array}\right.} \\ {f(-3), f(0), f(2), f(3), f(5)}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Evaluate at x=-3

Since \(-3\) is less than \(2\), we use the first rule of the piecewise function. Therefore, \(f(-3) = 5\).

Step 2: Evaluate at x=0

Since \(0\) is less than \(2\), we follow the first rule of the function as well. So, \(f(0) = 5\).

Step 3: Evaluate at x=2

Since \(2\) is exactly equal to \(2\), we still use the first part of the piecewise function. Hence, \(f(2) = 5\).

Step 4: Evaluate at x=3

Since \(3\) is greater than \(2\), we use the second rule, \(2x - 3\). Calculate \(f(3) = 2(3) - 3 = 6 - 3 = 3\).

Step 5: Evaluate at x=5

Since \(5\) is also greater than \(2\), apply the second rule of the function. Calculate \(f(5) = 2(5) - 3 = 10 - 3 = 7\).

Key Concepts

Evaluating Functions

Evaluating a function essentially means finding the value of the function for a given input. In the context of a piecewise function, this involves determining which part of the function applies to a specific value of the variable. When given a set of rules like in the original exercise, you need to assess the input and identify the relevant rule.

• First, determine the range or condition that your input falls into.
• Next, apply the function rule corresponding to that range.
• Finally, perform the arithmetic to find the value of the function.

This step-by-step process helps ensure that you are using the correct calculation method. Evaluating functions consistently and accurately is foundational in mathematics.

Function Rules

Function rules provide the instructions needed to calculate the function's output. Each piece of a piecewise function comes with its specific rule that applies under certain conditions. Understanding these rules involves several steps:

• Identify which mathematical operation you must perform.
• Pay attention to the conditions given for each rule. These conditions determine whether a rule is applicable for a specific input.
• Remember that multiple rules may exist, so being methodical is crucial to avoid mistakes.

In the given exercise, the rules are:1. For values where \(x \leq 2\), use \(f(x) = 5\).2. For values where \(x > 2\), apply \(f(x) = 2x - 3\).By following these rules precisely, you can accurately find the function's value for any input.

Piecewise Evaluation

Piecewise evaluation requires handling functions defined by multiple sub-functions, each with its condition. The key to mastering piecewise functions is understanding how to determine which rule applies at a given moment.
When evaluating piecewise functions:

• Start by identifying the input value and inspecting the conditions of all sub-functions.
• Match the condition with the correct piece of the function.
• Execute the operation specified by the appropriate rule.

For example, in our original exercise:- Values like \(x = -3, 0, 2\) fall under \(x \leq 2\) condition, so we use \(f(x) = 5\).- Values such as \(x = 3, 5\) fall under \(x > 2\), requiring us to calculate using \(f(x) = 2x - 3\).The ability to navigate these evaluations efficiently comes from practice and familiarity with piecewise function structures.

Mathematical Reasoning

Mathematical reasoning is the logical thought process we use to solve math problems correctly. For piecewise functions, it's essential not only to understand the rules but also to apply them carefully through reasoning.
To incorporate reasoning:

• Examine the structure of the piecewise function thoroughly.
• Break down the conditions and the calculations separately.
• Verify each step of evaluation to ensure the correct application of rules.

In this exercise, each function evaluation, such as \(f(-3) = 5\) or \(f(3) = 3\), reflects a decision reached through logical reasoning based on understanding the conditions attached to each rule.By employing mathematical reasoning alongside computation, you ensure that the function's evaluations are not only consistent but also rational and accurate. This skill is useful in more complex mathematical problems and helps build a strong analytical foundation.