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The verbal representation of a function means describing the steps you take to evaluate it using words instead of numbers or algebraic symbols. For the function given, the algebraic form is \[ f(x) = (x-4)^2 + 3 \] In words, you would describe this process sequentially. Firstly, you "subtract 4" from your input value, which is the number you are evaluating the function at. Next, "square the result," which means multiplying the resulting number by itself. Finally, "add 3" to the squared value. This sequence is a step-by-step oral or written instruction for how to generate the output from any input value.

The numerical representation of a function is shown through a table of input-output pairs. In our example, the task is to fill in the missing outputs for the function \[ f(x) = (x-4)^2 + 3 \] by substituting various values of \( x \) in the table. This representation shows exactly what number each input maps to as an output. - For \( x = 0 \), \( f(x) = (0-4)^2 + 3 = 19 \).- For \( x = 2 \), substitute into the function to get \( f(2) = (2-4)^2 + 3 = 7 \).- For \( x = 4 \), follow the calculation as \( f(4) = (4-4)^2 + 3 = 3 \).- Lastly, for \( x = 6 \), \( f(6) = (6-4)^2 + 3 = 7 \).These calculations complete the table and provide a clear picture of how the inputs relate to the outputs.

An algebraic representation of a function involves expressing the relationship between variables using algebraic symbols and operations. The function in our example is given as \[ f(x) = (x-4)^2 + 3 \]. In this form, the expression shows how the output, \( f(x) \), relates to the input, \( x \). - The operation \( (x-4) \) indicates a shift. You subtract 4 from \( x \), which is the base operation.- The square operation, \((x-4)^{2}\), represents a "spread" or modification that changes the shape of the graph taking inputs.- Adding 3 at the end, \(+ 3\), translates the output vertically in the graph by 3 units.This algebraic representation gives a comprehensive, symbolic description that is powerful and versatile, especially for graphing or further algebraic manipulation.

Function evaluation is the process of finding the output of a function for a particular input. In simple terms, when you "evaluate a function," you're plugging in a number into the function to see what result it gives back. Using the function \[ f(x) = (x-4)^2 + 3 \], let's see how you evaluate it:

• Choose a value for \(x\), such as \(x = 2\).
• Substitute into the function: \( f(2) = (2-4)^2 + 3\).
• Calculate: First, perform the inner operation \(2-4\) to get \(-2\).
• Next, square the result: \((-2)^2 = 4\).
• Finally, add 3 to obtain the output: \(4 + 3 = 7\).

This evaluation for different \(x\) values helps to verify the function's behavior and can be particularly useful when understanding how the function maps each input to its corresponding output.