Problem 19 - Algebra for College Students

Question

If \(f(x)=\left\\{\begin{array}{rll}1 & \text { for } x>0 \\ 0 & \text { for }-1

Step-by-Step Solution

Answer

\( f(2) = 1 \), \( f(0) = 0 \), \( f(-\frac{1}{2}) = 0 \), \( f(-4) = -1 \).

Step 1: Evaluate f(2)

Since 2 is greater than 0, we use the first part of the piecewise function: \( f(x) = 1 \) for \( x > 0 \). Therefore, \( f(2) = 1 \).

Step 2: Evaluate f(0)

0 is not greater than 0 but falls within the interval \(-1 < x \leq 0\). According to the piecewise function, \( f(x) = 0 \) for \(-1 < x \leq 0\). Hence, \( f(0) = 0 \).

Step 3: Evaluate f(-1/2)

\(-\frac{1}{2}\) is also in the interval \(-1 < x \leq 0\). So, according to the piecewise function, \( f(x) = 0 \) for \(-1 < x \leq 0\). Thus, \( f(-\frac{1}{2}) = 0 \).

Step 4: Evaluate f(-4)

Since -4 is less than -1, we use the last part of the piecewise function: \( f(x) = -1 \) for \( x \leq -1 \). Therefore, \( f(-4) = -1 \).

Key Concepts

Function Evaluation

Evaluating a piecewise function involves determining the output of the function based on specific intervals defined by the problem. In this exercise, we are provided with a piecewise function, which is a function defined by different expressions depending on which interval the input, or variable, falls into.
To evaluate the function, we compare the input value (or values of interest) with the conditions given for each piece of the function. For example, let's evaluate the function for different inputs:

• For input \( x = 2 \): This value is greater than 0, so we use the first piece of the function where the output is 1.
• For input \( x = 0 \): This value lies in the interval \(-1 < x \leq 0\), meaning we use the second piece where the output is 0.
• For input \( x = -\frac{1}{2} \): Similarly, this value is in the same interval as when \( x = 0 \), so the output is again 0.
• For input \( x = -4 \): Here, \( x \) is less than -1, so the last piece of the function tells us the output is -1.

Thus, evaluating a piecewise function requires careful attention to the intervals to determine which expression applies to the input value.

Mathematical Intervals

Mathematical intervals are ranges of numbers where certain conditions hold. In the context of this exercise, intervals help us determine which part of the piecewise function to use for evaluation.
The intervals specified for this problem are as follows:

• \( x > 0 \): This indicates any number greater than 0 falls in the first interval where the function outputs 1.
• \(-1 < x \leq 0 \): This interval includes numbers less than or equal to 0 but greater than -1. Within this range, the function provides an output of 0.
• \( x \leq -1 \): Finally, this covers all numbers less than or equal to -1, and the function gives an output of -1.

These intervals are crucial in piecewise function problems because they allow us to map specific input values to their corresponding outputs accurately based on the function's rules. Understanding how to read and apply intervals is essential in algebra and calculus.

Algebraic Solutions

Algebraic solutions involve solving equations or expressions using algebraic manipulations. With piecewise functions, solving involves determining how algebraic expressions provide outputs within specified conditions or intervals.
In this exercise, each piece of the function is simple and direct based on the specific algebraic expression provided:

• For \( x > 0 \): The function is simply \( f(x) = 1 \). Regardless of the value, as long as it's positive, the solution is straightforward.
• For \(-1 < x \leq 0 \): The expression \( f(x) = 0 \) means any value in this interval returns 0.
• For \( x \leq -1 \): Here, \( f(x) = -1 \) suggests that this part of the function outputs -1 for any value equal to or less than -1.

By understanding the algebraic side of piecewise functions, you can easily determine the output for any given input without getting confused by more complex equations. Each piece works independently within its interval, simplifying calculations if you follow the correct rules.