Problem 116 - Algebra and Trigonometry

Question

$$ \text { Let } f(x)=\left\\{\begin{aligned} x+3 & \text { if } x \geq 5 \\ 8 & \text { if } x<5 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The value of the function \(f(x)\) depends on the value of \(x\). If \(x\) is greater than or equal to 5, the value of \(f(x)\) is \(x + 3\). If \(x\) is less than 5, the value of \(f(x)\) is 8.

Step 1: Determine the value of x

In order to solve for \(f(x)\), it need to determined what value \(x\) has. If \(x\) is given as a specific number, use this number in the following steps. If \(x\) is not specified, the solution can be given in a general form for any given \(x\) based on the two different conditions in the piecewise function.

Step 2: Apply the correct condition

Once the value of \(x\) is known, apply the condition that corresponds to that value of \(x\). If \(x\) is greater than or equal to 5, use the function \(f(x) = x + 3\). If \(x\) is less than 5, use the function \(f(x) = 8\).

Step 3: Calculate the value of the function

After applying the correct condition, evaluate the function. If the function is \(f(x) = x + 3\), add \(x\) and 3. If the function is \(f(x) = 8\), the output value is simply 8.

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics, which involves determining the output of a function for a given input. It is like checking what comes out of a machine when you put something into it. For the piecewise function \( f(x) = \left\{ \begin{aligned} x+3 & \text{ if } x \geq 5 \ 8 & \text{ if } x < 5 \end{aligned} \right. \), evaluating the function depends on the value of \( x \). When evaluating functions, it is essential to choose the correct rule based on the value of \( x \):

• If \( x \geq 5 \), calculate \( f(x) \) using \( x+3 \).
• If \( x < 5 \), \( f(x) \) is simply 8, regardless of what \( x \) is.

To successfully evaluate a function, it is crucial to understand how to apply the right piece of the piecewise function depending on the input condition. This ensures correct and meaningful results.

Conditions of a Piecewise Function

Piecewise functions, as their name suggests, are defined by pieces. Each piece applies to a specific condition based on the input value. Understanding when and how to use each part of the function is fundamental to working with piecewise functions effectively.Here are the typical steps to consider conditions in a piecewise function:

• Identify the different conditions given in the problem. In the above function, the conditions are \( x \geq 5 \) and \( x < 5 \).
• For every input you want to evaluate, check which condition it satisfies. This will tell you which part of the function to use.
• Apply the rule that matches the condition. For example, if \( x = 7 \), since \( x \geq 5 \), the condition \( x+3 \) is applied, resulting in \( f(7) = 7 + 3 = 10 \).

By breaking down the pieces and understanding the conditions, tackling piecewise functions becomes much more manageable.

Mathematical Functions

Mathematical functions are an essential part of algebra and help express relationships between different quantities. A function relates an input to exactly one output. Knowing how to handle and interpret different types of functions expands your problem-solving toolbox. In any function, including piecewise functions, remember the following:

• Definition: It shows how each input value is transformed to an output with a specific rule.
• Types of Functions: Functions can come in many forms, like linear, quadratic, or, as in this case, piecewise, where there are different rules for different input ranges.
• Understanding the Function: Analyzing the conditions or domains where each part of the function applies is key to solving function-related problems.

By mastering the concept of functions, one develops a solid foundation that is applicable to advanced mathematical topics and real-world situations.