Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, given each function \(f,\) evaluate \(f(-3), \quad f(-2), \quad f(-1),\) and \(f(0) .\) \(f(x)=\left\\{\begin{array}{ll}1 & \text { if } x \leq-3 \\ 0 & \text { if } x>-3\end{array}\right.\)

Step-by-Step Solution

Answer

\( f(-3) = 1, \; f(-2) = 0, \; f(-1) = 0, \; f(0) = 0. \)

Step 1: Understanding the Function Definition

The function \( f(x) \) is a piecewise function. It is defined as 1 for all values of \( x \) that are less than or equal to \(-3\). For values of \( x \) greater than \(-3\), the function equals 0.

Step 2: Evaluating \( f(-3) \)

Since \( x = -3 \) is less than or equal to \(-3\), we use the first part of the piecewise function. Thus, \( f(-3) = 1 \).

Step 3: Evaluating \( f(-2) \)

For \( x = -2 \), which is greater than \(-3\), we apply the second part of the piecewise function. Therefore, \( f(-2) = 0 \).

Step 4: Evaluating \( f(-1) \)

Since \( x = -1 \) is greater than \(-3\), the second part of the function is applicable. Hence, \( f(-1) = 0 \).

Step 5: Evaluating \( f(0) \)

Here, \( x = 0 \) is also greater than \(-3\), hence the second condition of the function applies. Thus, \( f(0) = 0 \).

Key Concepts

Function Evaluation

In mathematics, function evaluation is the process of finding the output of a function given an input value. It's like following a recipe: you put in specific ingredients (the input values) and get out a dish (the function's output). Each function has its rule or rules, which tell you how to compute the output from the input. For example, take the function \( f(x) = x^2 \). To evaluate \( f(3) \), you substitute \( 3 \) into the function: \( f(3) = 3^2 = 9 \). So, the output when \( 3 \) is plugged into the function \( f(x) \) is \( 9 \).
Understanding function evaluation is crucial as it provides the foundation for analyzing more complex functions and solving equations that arise in many areas of mathematics and real-world problems.

Piecewise Function Definition

A piecewise function is a function that is defined by different expressions for different parts of its domain. Think of it like a "choose your own adventure" book, where the story changes depending on your choices. Piecewise functions are common in real-life scenarios where conditions change over time or depending on certain factors.

• Each "piece" of the function corresponds to a specific condition or range of the domain.
• The function can switch from one expression to another, based on the input value.

For instance, the function \( f(x) \) in the exercise is \( f(x)=\left\{\begin{array}{ll}1 & \text{if } x \leq -3 \ 0 & \text{if } x > -3\end{array}\right. \). This means if \( x \) is \(-3\) or less, the function returns \(1\), and any \( x \) greater than \(-3\) results in \(0\).
Understanding piecewise functions is essential since they can model complex situations more accurately than a single expression function.

Evaluating Piecewise Functions

When evaluating piecewise functions, it's important to identify which piece of the function to use based on the given input value. Think of it like navigating a map: you need to know which path to take based on your starting point. Here’s how you can evaluate a piecewise function:

• Examine the value you need to evaluate—let's call it \( x \).
• Check which condition \( x \) satisfies in the piecewise function.
• Use the appropriate formula or expression from the condition to find the value of the function at \( x \).

For instance, the function \( f(x) \) provided in the exercise has two conditions: if \( x \leq -3 \), we find that \( f(x) = 1 \); if \( x > -3 \), then \( f(x) = 0 \). Evaluating \( f(-3), f(-2), f(-1), \) and \( f(0) \) requires applying these steps to determine which part of the function applies to each value.
Mastering the evaluation of piecewise functions will aid in understanding diverse mathematical models and their applications.