Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the table. $$f(s)=\frac{|s-2|}{s-2}$$ $$\begin{array}{|l|l|l|l|l|l|} \hline s & 0 & 1 & \frac{3}{2} & \frac{5}{2} & 4 \\ \hline f(s) & & & & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The completed table is: \[\begin{array}{|l|l|l|l|l|l|} \hline s & 0 & 1 & \frac{3}{2} & \frac{5}{2} & 4 \ \hline f(s) & -1 & -1 & -1 & 1 & 1\\hline \end{array}\]

Step 1: Understand the concept of absolute value

Absolute value of a number is its distance from 0, which is always positive. So, if \(a>b\) then \(|b-a|=|a-b|\). Here, if \(s>2\), \(|s-2|\) is positive and same is the case when \(s<2\). But denominator \(s-2\) will be positive when \(s>2\) and negative when \(s<2\). Hence, resulting in value of the function as 1 when \(s>2\) and -1 when \(s<2\).

Step 2: Calculate the values of f(s) for given s

Now using the function \(f(s)=\frac{|s-2|}{s-2}\), and applying the concept from step 1, let's evaluate \(f(s)\) for each given value of \(s\) in the table.

Step 3: Plug s = 0 into the function

When \(s = 0\), on calculating we get \(f(0) = \frac{|0-2|}{0-2} = -1\)

Step 4: Plug s = 1 into the function

When \(s = 1\), we have \(f(1) = \frac{|1-2|}{1-2} = -1\)

Step 5: Plug s = 3/2 into the function

When \(s = 3/2\), we get \(f(3/2) = \frac{|3/2-2|}{3/2-2} = -1\)

Step 6: Plug s = 5/2 into the function

When \(s = 5/2\), we find \(f(5/2) = \frac{|5/2-2|}{5/2-2} = 1\)

Step 7: Plug s = 4 into the function

When \(s = 4\), we get \(f(4) = \frac{|4-2|}{4-2} = 1\)

Key Concepts

Piecewise Functions

A piecewise function is a type of function that has different expressions or rules for different sections of its domain. These sections are determined by specific intervals of the input variable. Rather than having a single equation for all possible inputs, a piecewise function can change its formula depending on the value of the input.

• This type of function is commonly divided into regions based on specific conditions, such as when the input variable is less than, greater than, or equal to certain values.
• Each piece of the function is defined on a certain interval, and together they cover the entire domain of the function.

In our example, the function \(f(s) = \frac{|s-2|}{s-2}\) is a piecewise function. Here's why:

• If \(s > 2\), both the absolute value \(|s-2|\) and the expression \(s-2\) are positive, leading to a fraction \(\frac{|s-2|}{s-2} = 1\).
• If \(s < 2\), \(s-2\) becomes negative while the numerator remains positive; thus, the function evaluates to \(-1\).
• When \(s = 2\), the denominator becomes zero, making the function undefined.

Function Evaluation

Function evaluation refers to the process of finding the value of a function at a particular input. It involves substituting a specific value into the function's formula and performing any necessary calculations. This concept is crucial for understanding how a function behaves at different points.
In our given function \(f(s) = \frac{|s-2|}{s-2}\), evaluating the function consists of solving it for different values of \(s\). Let's break down the steps:

• Substitute the specific value of \(s\) into the function.
• Calculate the absolute value to resolve \(|s-2|\).
• Perform the division by \(s-2\) to find the final function value.

For example:

• For \(s = 0\), substituting gives \(f(0) = \frac{|0-2|}{0-2}\), simplifying to \(-1\).
• For \(s = 1\), substituting results in \(f(1) = \frac{|1-2|}{1-2}\), also \(-1\).
• For \(s = 5/2\), we have \(f(5/2) = \frac{|5/2-2|}{5/2-2} = 1\).

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. They are fundamental in algebra when representing and analyzing relationships between quantities. Understanding algebraic expressions is essential for manipulating equations and solving problems in mathematics.
Key components of algebraic expressions include:

• **Variables:** Symbols used to represent unknown values, such as \(s\) in our function.
• **Constants:** Fixed numerical values within the expression, like \(2\) in \(s-2\).
• **Operators:** Mathematical symbols that indicate operations such as addition, subtraction, multiplication, and division.

In our exercise, the expression \(f(s) = \frac{|s-2|}{s-2}\) involves variable \(s\), and it showcases how algebraic expressions combine with the absolute value to create a piecewise function.

• The use of \(|s-2|\) demonstrates how algebraic expressions can be modified to become part of more complex functions.
• The expression clearly shows the transformation from an algebraic form to a piecewise rule governing the function's behavior.