Problem 67 - Elementary and Intermediate Algebra

Question

Complete each table. See Example 4. $$ f(t)=|t-2| $$ $$ \begin{array}{|r|l|} \hline t & f(t) \\ \hline-1.7 & \\ 0.9 & \\ 5.4 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

For \( t = -1.7 \), \( f(t) = 3.7 \); for \( t = 0.9 \), \( f(t) = 1.1 \); for \( t = 5.4 \), \( f(t) = 3.4 \).

Step 1: Identify the Absolute Value Function

The function given is \( f(t) = |t - 2| \), which means we take the absolute value of \( t - 2 \). The absolute value function makes negatives positive.

Step 2: Calculate \( f(t) \) for \( t = -1.7 \)

Substitute \( t = -1.7 \) into the function: \( f(-1.7) = |-1.7 - 2| = |-3.7| = 3.7 \). Thus, for \( t = -1.7 \), \( f(t) = 3.7 \).

Step 3: Calculate \( f(t) \) for \( t = 0.9 \)

Substitute \( t = 0.9 \) into the function: \( f(0.9) = |0.9 - 2| = |-1.1| = 1.1 \). Thus, for \( t = 0.9 \), \( f(t) = 1.1 \).

Step 4: Calculate \( f(t) \) for \( t = 5.4 \)

Substitute \( t = 5.4 \) into the function: \( f(5.4) = |5.4 - 2| = |3.4| = 3.4 \). Thus, for \( t = 5.4 \), \( f(t) = 3.4 \).

Key Concepts

Function Evaluation

Function evaluation is simply about substituting values into a function to find the output. When we have a function like \( f(t) = |t - 2| \), we're looking at how the function behaves for different inputs of \( t \). Here, \( |t - 2| \) represents the absolute difference between \( t \) and 2. Let's break down this further:

• Choose a value for \( t \).
• Substitute this value into the function to replace \( t \).
• Simplify the expression inside the absolute value.
• Apply the absolute value, which means converting any negative results into positive ones.

You can think of function evaluation as a step-by-step process. Ultimately, it helps to find out what the function outputs or calculates when you feed it a specific input.

Algebraic Expressions

Understanding algebraic expressions is key to handling functions like \( f(t) = |t - 2| \). An algebraic expression consists of numbers, variables, and operations like addition or subtraction. Here, \( t - 2 \) is the part of the expression inside the absolute value operator.Breaking it down:

• "\( t \)" is a variable representing any real number.
• "\( -2 \)" is a constant being subtracted from \( t \).

The subtraction itself might yield a positive or a negative result, depending on whether \( t \) is larger or smaller than 2. When evaluating such expressions, it's crucial to follow the order of operations:

• Simplify inside the parentheses or absolute values first.
• Then, apply the absolute value to ensure all results stay positive.

This methodical approach makes solving complex algebraic expressions more manageable.

Piecewise Functions

Piecewise functions are functions that have different expressions depending on the input variable's value. Although our original function \( f(t) = |t - 2| \) is not explicitly a piecewise function, it can be interpreted in this way due to the nature of absolute values:

• If \( t - 2 \geq 0 \), the function is simply \( t - 2 \).
• If \( t - 2 < 0 \), the function becomes \( -(t - 2) \).

Translating absolute value functions into piecewise form helps us understand how they behave differently depending on the input range.Consider the expression \(|t - 2|\) which changes based on if the result of \( t - 2 \) is positive or negative.This step often simplifies the process of graphing the function or analyzing it for various properties because each piece can be considered separately. Understanding piecewise functions can significantly clarify the behavior of more complex functions in algebra.