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When you are asked to evaluate a function, it means you must find the output of the function for specific input values. In mathematical terms, evaluating a function involves replacing the variable in the function's formula with a number from the domain, and then performing any necessary arithmetic to solve for the corresponding function value. In this exercise, we are working with a piecewise function, which is a type of function that has different expressions depending on the value of the input variable, \(x\). When evaluating a piecewise function, it’s essential to:

• Determine which piece of the function to use based on the given input value.
• Substitute the input value into the correct piece of the function.
• Calculate the result using basic arithmetic operations.

Understanding these steps will help you find the function's value accurately for any given input.

The function value refers to the result you obtain when you evaluate a function at a specific input. It's what comes out when you plug a number into the function equation. For a constant piece in a piecewise function, like \(f(x) = 5\) when \(x \leq 2\), the function value remains the same regardless of the input value, provided it satisfies the condition. In the provided example:

• The function value of \(f(-3)\), \(f(0)\), and \(f(2)\) is 5, because these inputs meet the condition \(x \leq 2\).
• Whereas for \(f(3)\) and \(f(5)\), we calculate based on the expression \(2x - 3\), which gives us 3 and 7, respectively.

The function value lets us see how the output varies or stays constant for different input values.

A piecewise defined function is a function that uses different expressions for different intervals of the input variable. These expressions are dictated by conditions defined in the function's rule. Piecewise functions are useful for representing real-life situations where a rule changes based on certain criteria. In our exercise, the piecewise defined function \(f(x)\) is defined as:

• \(f(x) = 5\) when \(x \leq 2\).
• \(f(x) = 2x - 3\) when \(x > 2\).

This means when \(x\) is less than or equal to 2, the function acts like a constant function (outputting 5). However, for \(x\) values greater than 2, the function behaves like a linear function (output increasing with \(x\)). Understanding these characteristics allows us to know which rule to apply based on the input and predict the function's output logically for any specific value of \(x\).