Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-2: If possible, find (a) \(f^{-1}(5)\) and (b) \(g^{-1}(6)\) $$ \begin{aligned} &\begin{array}{|l|l|l|l|} \hline t & 0 & 3 & 5 \\ \hline f(t) & 2 & 5 & 6 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \boldsymbol{t} & 1 & 2 & 4 \\ \hline \boldsymbol{g}(t) & 3 & 6 & 6 \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

(a) \(f^{-1}(5) = 3\) (b) \(g^{-1}(6) = \{2, 4\}\)

Step 1: Understanding Inverse Functions

To find the inverse function value, \(f^{-1}(y)\), we need to locate the input \(t\) for which the function \(f(t)\) is equal to the given output \(y\). So, if \(f(t) = y\), then \(f^{-1}(y) = t\). Similarly, for \(g^{-1}(y)\).

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

Given the table for \(f(t)\): \(f(t)\) is 5 when \(t = 3\). Hence, \(f^{-1}(5) = 3\).

Step 3: Solve for \(g^{-1}(6)\)

In the table for \(g(t)\), \(g(t)\) is 6 for both \(t=2\) and \(t=4\). Thus, the inverse \(g^{-1}(6)\) can be either 2 or 4 because both inputs give the output 6.

Key Concepts

Function Tables

Function tables are a great way to visually understand functions and their corresponding outputs. They represent the relationship between inputs and outputs in an organized manner, usually displayed in rows and columns.
Each row typically consists of input values on one side and the corresponding function output values on the other side. By examining these tables, you can easily determine the output of a function for a given input without needing to perform any calculations.
Function tables help in simplifying the process of finding inverse values, as you can conveniently look up values that reverse the function's effect.

• The column with inputs generally represents the "domain" of the function.
• The column with outputs corresponds to the "range" of the function.

Finding Inverse Values

Finding inverse values can seem tricky at first, but function tables make this task much simpler. An inverse function essentially undoes the action of the original function.
When you have a table of values, finding the inverse of a given function output involves locating the relevant output in the table and then identifying the corresponding input. This reversed process allows you to find the input value, which generates that particular output in the function.
For example, if you are given an output from the function and want to find its inverse, follow these steps:

• First, locate the output value in the function's column.
• Look across to see which input value gave that output as per the function's definition.
• That input is your inverse value of the given output.

Let's consider a practical scenario: If you want to find \(f^{-1}(5)\), locate '5' in the output of function \(f(t)\), then find '3' which is the corresponding input. Therefore, \(f^{-1}(5) = 3\). If you wish to find \(g^{-1}(6)\), note that '6' occurs for both inputs 2 and 4 in \(g(t)\), meaning \(g^{-1}(6) = 2\) or \(4\).

Algebra Problem-Solving

Algebra problem-solving can initially feel like solving a puzzle, but it becomes manageable as you grasp the underlying concepts like inverse functions. The concept of inverse functions is an essential algebraic tool that helps us solve equations where we seek original inputs that yield given outputs.
When we deal with inverse functions, we focus on reversing the steps of a function's input-output process. A strong grasp of inverse functions broadens our algebraic problem-solving abilities greatly.

• Algebra uses inverse functions to break down complex equation solving into more manageable steps.
• In real-world scenarios, inverse functions can be used to track back from outcomes to sources or origins.

By practicing with function tables and understanding how to find inverse values, students can strengthen their problem-solving skills, enabling them to tackle more challenging algebra problems in the future.