Problem 55 - A Graphical Approach to College Algebra

Question

The graphing calculator screen on the left shows three functions: \(\mathrm{Y}_{1}, \mathrm{Y}_{2},\) and \(\mathrm{Y}_{3} .\) The last of these, \(\mathrm{Y}_{3}\), is defined as \(\mathrm{Y}_{1} \circ \mathrm{Y}_{2},\) indicated by the notation \(\mathrm{Y}_{3}=\mathrm{Y}_{1}\left(\mathrm{Y}_{2}\right) .\) The table on the right shows selected values of \(\mathbf{X},\) along with the calculated values of \(\mathbf{Y}_{3} .\) Predict the display for \(\mathbf{Y}_{3}\) for the given value of \(\mathbf{X}\). $$\mathbf{X}=-1$$

Step-by-Step Solution

Answer

Substitute \(-1\) into \(Y_2(x)\), use result in \(Y_1(x)\) to find \(Y_3(-1)\).

Step 1: Understand the Composition Function

The function \(Y_3=X(Y_2)\) is the composition of \(Y_1\) and \(Y_2\). Composition means substituting \(Y_2(x)\) into \(Y_1\). Thus, \(Y_3(x) = Y_1(Y_2(x))\). We will calculate \(Y_3\) by finding \(Y_2(x)\) first and using it to find \(Y_1\).

Step 2: Find \(Y_2(-1)\)

Based on the values for \(X\) from the table, substitute \(X = -1\) into \(Y_2(x)\) to calculate the output. Let's assume \(Y_2(x) = ax + b\). Insert \(X = -1\) into this function to find \(Y_2(-1)\).

Step 3: Calculate \(Y_1(Y_2(-1))\) to find \(Y_3(-1)\)

After finding \(Y_2(-1)\), use this result as the input for the function \(Y_1(x)\). Find \(Y_1(Y_2(-1))\) which gives \(Y_3(-1)\). As an example, if \(Y_1(x) = cx + d\), simply substitute \(Y_2(-1)\) as the new \(x\).

Step 4: Final Step: Verify Output

Match your result for \(Y_3(-1)\) with the pattern of values in the table to confirm your prediction is accurate.

Key Concepts

Using a Graphing Calculator Efficiently

Graphing calculators are invaluable tools for solving functions and their compositions. They help us visualize functions by providing a graphical representation. For the function composition problem presented, a graphing calculator displays the output of three functions: \(Y_1\), \(Y_2\), and their composition \(Y_3 = Y_1(Y_2)\). By entering these functions, you can easily see how changes to \(X\) affect \(Y_3\).
To predict values efficiently:

• Enter the function expressions into the calculator using the appropriate function keys.
• Make sure the calculator is set to the correct mode (i.e., function mode).
• Input different \(X\) values to see the resulting \(Y_3\) values.

Utilizing a graphing calculator allows you to analyze whether your manual compositions align with the real-time graphical data provided. Thus, verifying predictions becomes far simpler!
Use this visual aid selected values of \(X\), make predictions about \(Y_3\), and confirm them by entering them into your calculator.

Understanding Function Notation

Function notation is a concise way of representing mathematical functions and their operations, which is crucial in understanding function compositions. When you see an expression like \(Y_3 = Y_1(Y_2)\), it signifies that \(Y_2\) is an input into \(Y_1\). Think of this as fitting one function's output into another function's input.
This can be visualized as a chain of operations:

• First, evaluate \(Y_2\) for a particular \(X\) to get the intermediate result.
• Then, substitute this result into \(Y_1(x)\) to calculate \(Y_3(x)\).

Function notation simplifies complex operations, allowing you to clearly see the sequence and interrelation of functions involved. Comprehending this notation makes understanding and predicting outcomes of composition more straightforward.

Mastering Composition Function Calculation

Understanding how to calculate composition functions is essential for tackling problems involving these mathematical operations. When you have \(Y_1(Y_2(x))\), you're essentially processing function \(Y_2\) first, getting an output, and using this output as an input for \(Y_1\).
Let's break it down with the given exercise:

• Step 1: Calculate \(Y_2(-1)\). Suppose \(Y_2(x) = ax + b\). By substituting \(x = -1\), you'll find \(Y_2(-1)\).
• Step 2: Use \(Y_2(-1)\) as the new input for \(Y_1(x)\). If \(Y_1(x) = cx + d\), substitute \(Y_2(-1)\) into it to calculate \(Y_1(Y_2(-1))\).
• Step 3: This result gives you \(Y_3(-1)\).

By following these steps, you ensure that all calculations are sequential and accurate. Verify your results against known values for further confirmation. Practicing this process enhances your confidence and ability to handle complex compositions.