Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-2: If possible, find (a) \(f^{-1}(5)\) and (b) \(g^{-1}(6)\) $$ \begin{aligned} &\begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & 2 & 4 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 3 & 5 & 9 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline x & 1 & 3 & 5 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & 6 & 2 & 6 \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

(a) \(f^{-1}(5) = 4\); (b) \(g^{-1}(6) = 1\) or \(5\).

Step 1: Understanding the Inverse Function

The inverse function, \(f^{-1}(y)\), is found by determining which input \(x\) to a function \(f(x)\) gives the output \(y\). In a table, look for the output value \(y\) and use the corresponding input as the inverse.

Step 2: Finding \(f^{-1}(5)\)

Refer to the table for the function \(f(x)\). We need to find the \(x\) value such that \(f(x) = 5\). From the provided table, \(f(4) = 5\). Therefore, \(f^{-1}(5) = 4\).

Step 3: Checking the Function \(g(x)\)

To find \(g^{-1}(6)\), check the table for \(g(x)\) to find the input \(x\) that gives the output \(g(x) = 6\).

Step 4: Finding \(g^{-1}(6)\)

In the table for \(g(x)\), both \(x = 1\) and \(x = 5\) give \(g(x) = 6\). Thus, \(g^{-1}(6) = 1 \text{ or } 5\).

Key Concepts

Function Table

A function table is a simple yet powerful way to visualize the relationship between inputs and outputs of a function. In such tables, each row corresponds to an input-output pair, neatly organizing this information for easy reference. These tables often have two columns:

• The first column lists possible values of the independent variable, often denoted as \( x \).
• The second column contains the corresponding values of the function output, denoted as \( f(x) \) or \( g(x) \) depending on the function in question.

Each entry or row in the table shows one complete input-output pair. This systematic arrangement assists us in analyzing how changing one variable affects the other. For problems involving inverse functions, function tables help us trace the output values back to their original inputs, reversing the direction of examination compared to when we're simply evaluating the function.

Input-Output Pairs

Input-output pairs are the core building blocks of understanding any function and, by extension, its inverse. Each pair consists of an input \( x \) and its corresponding output \( y = f(x) \). These pairs establish a direct map from the domain (all possible \( x \)-values) to the range (all possible \( y \)-values) of a function.A critical insight when dealing with input-output pairs is their ability to directly inform inverse functions. To find an inverse, you need to look at these pairs in reverse. For instance, finding \( f^{-1}(5) \) involves locating the pair where the output equals 5 and then determining the original input value, which is what the inverse function returns.For the function \( g(x) \), the task required finding \( g^{-1}(6) \). This solution features another important aspect: sometimes, multiple inputs may lead to the same output, indicating that the inverse might not be a function or could have multiple values. In such cases, as seen with \( g^{-1}(6) = 1 \text{ or } 5 \), both inputs provide valid answers, highlighting the richness of analyzing input-output pairs.

Inverse Operations

Inverse operations are a key concept in algebra, often used to "undo" a given operation. When it comes to inverse functions, the idea is similar. An inverse function essentially reverses the effect of the original function. If you think of a function as moving from input to output, the inverse function takes you from output back to input.Understanding how to derive or interpret an inverse operation requires following these steps:

• Identify the output value for which you're finding the inverse.
• Search through the function table to locate this output.
• Trace back to find which input produced this output, thereby executing the inverse operation.

For example, when we determined that \( f^{-1}(5) = 4 \), we successfully used the inverse operation to find which input led to the output of 5. In time and with practice, recognizing and applying inverse operations using tables will become second nature, illuminating deeper understanding as functions intertwine through real-life problems.