91Ó°ÊÓ

In mathematics, the absolute value of a number is a fundamental concept. It represents the magnitude of a number, disregarding its sign. This means it tells us how "far" a number is from zero, whether it's positive or negative. For example, both the numbers "+3" and "-3" have an absolute value of "3".
This concept becomes essential when dealing with equations or functions that involve distance or magnitude.

• Absolute value is denoted by vertical bars, like so: \(|x|\).
• For any positive number or zero, the absolute value remains the same.
• For any negative number, the absolute value is its positive counterpart.

In the function given, \( f(T) = 7.2 - 2.5|T| \), the absolute value of \( T \) ensures that \( f(T) \) responds to the size of \( T \), leveraging the magnitude while ignoring the sign. This greatly simplifies calculations when it's necessary to consider only the size of the input value.

Linear functions are one of the most straightforward yet powerful tools in mathematics. They describe a relationship where there is a constant rate of change.
In its simplest form, a linear function can be expressed as \( y = mx + b \), where:

• \( m \) represents the slope or rate of change.
• \( b \) is the y-intercept, where the line crosses the y-axis.

The function \( f(T) = 7.2 - 2.5|T| \) is slightly more complex due to the presence of the absolute value. However, it remains linear relative to the usage of \( T \). In such functions, values calculated after evaluating the absolute term change linearly.
The role of the slope (-2.5 in this case) tells us how steeply the function decreases per unit increase in \( |T| \). This demonstrates that for each unit increase in \( |T| \), the function value decreases by \( 2.5 \).

The substitution method is a straightforward technique for function evaluation. It involves directly replacing the variable in a function with a specific value to find the output.
To apply the substitution method, remember these steps:

• Identify the function and the specific value you want to substitute.
• Replace the variable inside the function with the given value.
• Simplify the expression to find the result.

In this exercise, we substituted \( T = 2.6 \) and then \( T = -4 \) into the function \( f(T) = 7.2 - 2.5|T| \). Using the substitution method here means replacing \( T \) in the equation with the specified numbers, computing the absolute value, and then simplifying to obtain \( f(2.6) = 0.7 \) and \( f(-4) = -2.8 \). This method simplifies the process of evaluating even more complex functions efficiently.