Problem 60 - Algebra 1

Question

EVALUATING FUNCTIONS Evaluate the function for these values of \(x:-2,-1,0,\) and \(1 .\) Organize your results in a table. $$ y=12-x $$

Step-by-Step Solution

Answer

The evaluated result of the function \(y = 12 - x\) for the specific values of \(x\) are as follows: for \(x = -2\), \(y = 14\); for \(x = -1\), \(y = 13\); for \(x = 0\), \(y = 12\); for \(x = 1\), \(y = 11\). These values can be tabulated in the following way: {(-2,14),(-1,13),(0,12),(1,11)}.

Step 1: Substitution of \(x = -2\)

Substitute \(x = -2\) into the equation \(y = 12 - x\). Therefore, \(y = 12 - (-2) = 12 + 2 = 14\).

Step 2: Substitution of \(x = -1\)

Substitute \(x = -1\) into the equation \(y = 12 - x\). Therefore, \(y = 12 - (-1) = 12 + 1 = 13\).

Step 3: Substitution of \(x = 0\)

Substitute \(x = 0\) into the equation \(y = 12 - x\). Therefore, \(y = 12 - 0 = 12\).

Step 4: Substitution of \(x = 1\)

Substitute \(x = 1\) into the equation \(y = 12 - x\). Therefore, \(y = 12 - 1 = 11\).

Step 5: Organizing the results in a table

Organize the calculated values of \(y\) for each value of \(x\) into a table. The table will have two columns, one for the \(x\) values and one for the corresponding \(y\) values. The table data will be as follows: \(x = -2, y = 14\); \(x = -1, y = 13\); \(x = 0, y = 12\); \(x = 1, y = 11\).

Key Concepts

Substitution Method

When evaluating functions, the substitution method is a fundamental technique used to find the output value for a given input. Essentially, it involves replacing the variable in the function with the provided number or expression. In our exercise, we are given the function y = 12 - x and we're asked to evaluate it for various values of x: -2, -1, 0, and 1.

For instance, when we substitute x = -2 into the function, we replace every instance of x in the equation with -2, leading us to calculate y = 12 - (-2) = 14. This process is repeated for each value of x. Remember to follow the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure accuracy in your results.

• For negative numbers, the subtraction turns into addition because subtracting a negative is the same as adding the positive equivalent of that number.
• When substituting zero, the term including zero often eliminates itself, simplifying the evaluation.
• Keep special attention to positive numbers to ensure you perform the correct arithmetic operation as specified in the function.

Function Evaluation

Evaluating a function is like using a very specific recipe for your math expressions: the function tells you what to do with the ingredients (the input values), and you end up with a finished product (the output value). Imagine your function y=12-x as instructions for baking a cake where x is how much sugar to take away from a pre-set amount.

Each time you change the amount of sugar (swap out different values of x), the outcome of your cake (the value of y) is different. That's essentially what you're doing in function evaluation – you see how changing the input affects the output.

• Be thorough with each step to avoid mixing up the values, especially when dealing with negative numbers.
• Always substitute the values carefully to avoid any calculation mistakes.
• Double-check your results for each calculation to ensure that it aligns with the mathematical rules.

By following these steps meticulously, you help ensure that your function evaluation process yields accurate and reliable results.

Organizing Data in a Table

Once you've calculated the different values of y for each x in your function, it's time to organize the data in a table. This helps provide a clear visual representation of how the input values correspond to the output values. In our function evaluation exercise, having a two-column table is the most straightforward approach: one column for x (the inputs), and another for y (the outputs).

By organizing the data this way:

• You create a visual reference that makes the relationship between x and y easy to understand.
• It becomes simpler to identify any patterns in the way the function behaves with different inputs.
• You lay the foundation for more complex analysis, such as graphical representation or statistical calculations.

Remember to align your x values with the correct y values, and consider using a ruler or digital spreadsheet to keep your table tidy and accurate. An organized table ensures that the evaluation of the function is easily understood and the data is ready for further analysis if necessary.