Problem 23 - Algebra and Trigonometry

Question

Evaluate the piece wise defined function at the indicated values. $$ f(x)=\left\\{\begin{array}{l}{x^{2}+2 x} \\ {x} \\ {-1}\end{array}\right. $$ $$ \begin{array}{l}{\text { if } x \leq -1} \\ {\text { if }-1< x \leq 1} \\\ {\text { if } x >1}\end{array} $$ $$ f(-4), f\left(-\frac{3}{2}\right), f(-1), f(0), f(25) $$

Step-by-Step Solution

Answer

\( f(-4)=8, f\left(-\frac{3}{2}\right)=-\frac{3}{4}, f(-1)=-1, f(0)=0, f(25)=-1 \)

Step 1: Understanding the Piecewise Function

The function is defined in three different ways depending on the value of \( x \):1. \( f(x) = x^2 + 2x \) if \( x \leq -1 \).2. \( f(x) = x \) if \( -1 < x \leq 1 \).3. \( f(x) = -1 \) if \( x > 1 \).

Step 2: Evaluating at \( x = -4 \)

Since \( -4 \leq -1 \), we use \( f(x) = x^2 + 2x \).Thus, \( f(-4) = (-4)^2 + 2(-4) = 16 - 8 = 8 \).

Step 3: Evaluating at \( x = -\frac{3}{2} \)

Since \( -\frac{3}{2} \leq -1 \), we use \( f(x) = x^2 + 2x \).Thus, \( f\left(-\frac{3}{2}\right) = \left(-\frac{3}{2}\right)^2 + 2\left(-\frac{3}{2}\right) = \frac{9}{4} - 3 = -\frac{3}{4} \).

Step 4: Evaluating at \( x = -1 \)

Since \( x = -1 \leq -1 \), we use \( f(x) = x^2 + 2x \).Thus, \( f(-1) = (-1)^2 + 2(-1) = 1 - 2 = -1 \).

Step 5: Evaluating at \( x = 0 \)

Since \( -1 < 0 \leq 1 \), we use \( f(x) = x \).Thus, \( f(0) = 0 \).

Step 6: Evaluating at \( x = 25 \)

Since \( x > 1 \), we use \( f(x) = -1 \).Thus, \( f(25) = -1 \).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input value, using the rules or formulas defined by the function. In our exercise, we're dealing with a piecewise function, which means that its formula or rule changes depending on the value of the input \( x \).

To evaluate this function, you must first determine which condition \( x \) satisfies and then use the corresponding part of the piecewise function definition. This involves substituting the \( x \) value into the appropriate expression to find \( f(x) \). Here’s what you should keep in mind for evaluating a piecewise function:

• Check against each condition sequentially.
• Determine which condition \( x \) satisfies.
• Use the formula specific to that condition to calculate \( f(x) \).

Through this method, the unique properties of piecewise functions are utilized to find accurate outputs based on their defined sections for different inputs.

Step-by-Step Solution

A step-by-step solution helps in systematically breaking down the evaluation process of a piecewise function as shown in the original solution. The process ensures accuracy and aids in understanding.

1. **Evaluating at \( x = -4 \):**
Identify that \( -4 \leq -1 \). Hence, use \( f(x) = x^2 + 2x \). After substituting \( x = -4 \), calculate the result: \[ f(-4) = (-4)^2 + 2(-4) = 16 - 8 = 8 \]

2. **Evaluating at \( x = -\frac{3}{2} \):**
The condition \( -\frac{3}{2} \leq -1 \) indicates the same formula as above. Substituting, the calculation yields: \[ f\left(-\frac{3}{2}\right) = \left(-\frac{3}{2}\right)^2 + 2\left(-\frac{3}{2}\right) = \frac{9}{4} - 3 = -\frac{3}{4} \]

3. **Evaluating at \( x = -1 \):**
Since \( x = -1 \leq -1 \), again use \( f(x) = x^2 + 2x \), giving: \[ f(-1) = (-1)^2 + 2(-1) = 1 - 2 = -1 \]

4. **Evaluating at \( x = 0 \):**
Here, \( -1 < 0 \leq 1 \) points to \( f(x) = x \). Thus:\[ f(0) = 0 \]

5. **Evaluating at \( x = 25 \):**
Since \( x > 1 \), apply \( f(x) = -1 \), resulting directly in:\[ f(25) = -1 \]

Following a structured approach like this ensures no steps are overlooked, making sure the final outcomes are correct.

Piecewise Definition

A piecewise function is a function composed of multiple "pieces," each defined by its own formula and applicable on specific intervals of the input value \( x \). This type of function allows for different rules to apply in different situations, offering flexibility where a single rule wouldn’t suffice.

In this exercise, the function \( f(x) \) is broken down into three pieces:

• \( f(x) = x^2 + 2x \) when \( x \leq -1 \)
• \( f(x) = x \) when \( -1 < x \leq 1 \)
• \( f(x) = -1 \) when \( x > 1 \)

The conditions and domains are written explicitly along with each formula, specifying clearly which rule of the function applies for which range of values.

This structured definition can represent phenomena that exhibit distinct behaviors across different circumstances, such as different tax brackets in finance, or piecewise linear graphs. By understanding piecewise functions, you gain the ability to explore and analyze complex functions that express real-world situations more accurately.