Problem 31 - Algebra for College Students

Question

\(f(x)=\left\\{\begin{aligned} 2 & \text { for } x \geq 0 \\\\-1 & \text { for } x<0 \end{aligned}\right.\)

Step-by-Step Solution

Answer

The function outputs 2 for \(x \geq 0\) and -1 for \(x < 0\).

Step 1: Identify the Function

The function given is a piecewise function. It is defined differently for different intervals of the domain: for \(x \geq 0\), \(f(x) = 2\), and for \(x < 0\), \(f(x) = -1\).

Step 2: Determine the Value for \(x \geq 0\)

For any value of \(x\) that is greater than or equal to zero, the function takes the value of 2. For example, if \(x = 0\), then \(f(0) = 2\), or if \(x = 5\), then \(f(5) = 2\).

Step 3: Determine the Value for \(x < 0\)

For any value of \(x\) that is less than zero, the function takes the value of -1. For instance, if \(x = -1\), then \(f(-1) = -1\), or if \(x = -10\), then \(f(-10) = -1\).

Key Concepts

Function Evaluation

Evaluating a function means finding the output or the value of the function for a given input. In the case of piecewise functions, we need to be extra careful because the function can have different expressions based on the input. To evaluate our specific piecewise function, determine which part of the function you need to use based on the value of the input, \( x \).

• For inputs where \( x \geq 0 \), select the first part of the function, \( f(x) = 2 \).
• For inputs where \( x < 0 \), use the second part, \( f(x) = -1 \).

For example, if we evaluate \( f(3) \), since \( 3 \geq 0 \), we get \( f(3) = 2 \). If instead, we evaluate \( f(-5) \), since \( -5 < 0 \), we get \( f(-5) = -1 \). Function evaluation is about plugging in the input value into the correct section of the function to obtain the result.

Domain and Range

When dealing with piecewise functions, understanding the domain and range is crucial. The domain essentially refers to all possible input values \( x \) that the function can accept. In our function, the domain includes all real numbers because every real \( x \) value fits into one of the defined intervals:

• "\( x \geq 0 \)" encompasses zero and all positive numbers.
• "\( x < 0 \)" encompasses all negative numbers.

The range pertains to all possible output values of the function. Since our function has two distinct constant values:

• Whenever \( x \geq 0 \), the output is 2,
• and whenever \( x < 0 \), the output is -1.

Thus, the range of this function is \{2, -1\}. In essence, although the input can be any real number, the output is restricted to only these two distinct values.

Function Definition

A piecewise function is defined by different expressions for different parts of its domain. This means that the function's rule or equation changes based on the input intervals. For a function like \( f(x) = \begin{cases} 2 & \text{if } x \geq 0 \ -1 & \text{if } x < 0 \end{cases}\), we have two distinct behavior patterns defined:

• \( f(x) = 2 \) for all \( x \geq 0 \), indicating that this part of the function is constant at 2 when \( x \) fits this condition.
• \( f(x) = -1 \) for all \( x < 0 \), meaning the value is fixed at -1 in this interval.

Defining a piecewise function requires clearly specifying the condition and the corresponding output for each segment. It is crucial to note where each piece applies to avoid confusion when evaluating inputs that lie on the border between two parts, such as \( x = 0 \) in our example.