Problem 16 - The Calculus with Analytic Geometry

Question

\(f(x)=\left\\{\begin{aligned} 5-2 x & \text { if } x<3 \\ 3 x-10 & \text { if } 3 \leq x \end{aligned}\right.\)

Step-by-Step Solution

Answer

For \(x = 2\), \(f(2) = 1\). For \(x \geq 3\), \(f(4)=2\).

Step 1: Analyze the given piecewise function

The function is given as: \[ f(x) = \begin{cases} 5 - 2x & \text{if } x < 3 \ 3x - 10 & \text{if } x \geq 3 \end{cases} \] This indicates that the function has different expressions based on the value of \(x\). When \(x < 3\), use the equation \(5 - 2x\). When \(x \geq 3\), use the equation \(3x - 10\).

Step 2: Evaluate the function at specific points

Choose points for evaluation from both domains. For example: - To find \(f(2)\), since \(2 < 3\), use \(5 - 2(2)\). - To find \(f(4)\), since \(4 \geq 3\), use \(3(4) - 10\).

Step 3: Calculate using the appropriate expressions

Substitute the values: - For \(f(2)\), calculate: \[ f(2) = 5 - 2(2) = 5 - 4 = 1 \] - For \(f(4)\), calculate: \[ f(4) = 3(4) - 10 = 12 - 10 = 2 \]

Step 4: Summarize the findings

Using the piecewise function, we evaluated two points: - \(f(2) = 1\) - \(f(4) = 2\)

Key Concepts

Function Evaluation

Understanding piecewise functions starts with function evaluation. A piecewise function is defined by different expressions for different ranges of the independent variable, typically denoted as x.
In the provided exercise, the function is:
\[ f(x) = \begin{cases} 5 - 2x & \text{if } x < 3 \ 3x - 10 & \text{if } x \geq 3 \end{cases} \]
This means:

• Use the formula \(5 - 2x\) when \(x\) is less than 3.
• Use the formula \(3x - 10\) when \(x\) is 3 or greater.

Function evaluation involves determining which part of the piecewise function to use based on the value of x. The key is to carefully assess which condition x fulfills and then apply the corresponding equation. This ensures accurate calculation of function values for specific points.

Domain-Specific Calculations

Calculating values for piecewise functions involves domain-specific processes where you must match conditions to the correct expressions. Let's break it down using the given function.

When we need to find \(f(2)\):

• First, identify which part of the function to use. Since \(2 < 3\), we use the expression \(5 - 2x\).
• Substitute \(x = 2\) into the expression: \[ f(2) = 5 - 2(2) \]
• Solve the equation: \[ f(2) = 5 - 4 = 1 \]

Similarly, for \(f(4)\):

• Since \(4 \geq 3\), use the expression \(3x - 10\).
• Substitute \(x = 4\) into the expression: \[ f(4) = 3(4) - 10 \]
• Solve the equation: \[ f(4) = 12 - 10 = 2 \]

These simple steps ensure correct domain-specific calculations, making sure each value is correctly evaluated according to its appropriate function range.

Step-by-Step Solution

Let's walk through the solution using a step-by-step approach to make things clearer:

Step 1: Analyze the given piecewise function to understand its structure. The function is:
\[ f(x) = \begin{cases} 5 - 2x & \text{if } x < 3 \ 3x - 10 & \text{if } x \geq 3 \end{cases} \]

Step 2: Choose points for evaluation to cover different parts of the function. For instance, pick \(x = 2\) and \(x = 4\).

• To find \(f(2)\), since \(2 < 3\), use \(5 - 2(2)\).
• To find \(f(4)\), since \(4 \geq 3\), use \(3(4) - 10\).

Step 3: Calculate using the appropriate expressions.

• For \(f(2)\), substitute \(x = 2\): \[ f(2) = 5 - 2(2) = 1 \]
• For \(f(4)\), substitute \(x = 4\): \[ f(4) = 3(4) - 10 = 2 \]

Phew! We just evaluated the function at two different points. This is critical for understanding how piecewise functions change behavior depending on the values of x.

Step 4: Summarize the findings. Using the piecewise function, we've determined:

• \(f(2) = 1\)
• \(f(4) = 2\)

Simplified approaches like this ensure a strong grasp of evaluating piecewise functions. Keep practicing these steps to master the concept!