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Function evaluation is the process of determining the output of a function for a given input. It's like feeding the function a certain value and asking it, "What do you give me in return?" To find the value of a function, you simply substitute the provided variable into the function's equation.
For example, with our exercise, we need to determine \( f(T)=7.2-2.5|T| \) for specific values of \( T \). This involves replacing the \( T \) in the function with the given numbers, such as 2.6 and -4 in our case. Once you substitute, you simply follow the order of operations to solve the expression.
The absolute value function involved here is crucial because it converts any negative input into a positive one, ensuring you always work with non-negative values.

Mathematical functions are like machines that take an input and provide an output according to specific rules. Each function has a formula that dictates how it responds to any given input.
The function \( f(T) = 7.2 - 2.5|T| \) is an example of an absolute value function. Absolute value functions are a special kind of function since they involve the absolute value operation.

• Absolute value, represented by bars like this \(|T|\), outputs the distance of a number from zero, always resulting in a non-negative value.
• In our function, \( 7.2 \) is the constant term, and \(-2.5|T|\) represents a linear relationship with \( T \). Multiplying the absolute value with -2.5 scales the input but also inverts its impact, as shown by how outputs decrease or become negative in this context.

Step-by-step solutions make mathematical problems much easier to solve by breaking them down into smaller, manageable parts. Let's walk through our exercise step by step.

• Start by understanding the function and identifying what needs to be calculated. In our case, we need \( f(T) \) for \( T=2.6 \) and \( T=-4 \).
• Substitute each value into the function. Replace \( T \) with the given number, taking care of absolute values correctly. In this exercise, \(|2.6|=2.6\), and \(|-4|=4\).
• Perform the arithmetic step to simplify the expressions. For \( f(2.6) \), it's \( 7.2 - 6.5 = 0.7 \). For \( f(-4) \), it's \( 7.2 - 10 = -2.8 \).

This method ensures precision while instilling confidence in problem-solving. Remember, practice is key, and following this structured approach helps in mastering even the most complex problems.